What about motivating intuitions? Topology was developed basically to deal with intuitions about "space," "connectivity, "continuity," notions of "near" and "far," etc. Algebra came about in order to deal with notions of "finitary manipulation," especially in connection with equalities. The historical developments happened quite organically, without any particular "guiding hand," so the boundaries are fuzzy indeed...

Dear Demetris,

In mathematics a general structure is a system (X, R, F, C), where X is a non empty set, R is a family of relations, F is a family of operations, and C is a family of distinguished constants. For a topological space (X,τ), we can write each open set G as a unary relation, R_G. Thus a topological space is a relational structure (X, (R_G), G\in τ). A topological structure is essentially dealing with “closeness”, limits, etc. Using Nonstandard analysis one can regard topological-Analysis structures as algebraic structures. Thus we may say that everything, in concept is algebra. however the intension of each structure is quite different.

http://en.wikipedia.org/wiki/Kuratowski_closure_axioms

http://en.wikipedia.org/wiki/Closure_operator

See also,

TOPOLOGICAL BOOLEAN ALGEBRAS, Ch 3 in Rasiowa-Sikorski, The Metamathematics of Mathematics.

Notice that closure or interior operators are defined on the power Boolean Algebra, P(X). So that we have a topological Boolean algebra. The closure operator is just another algebraic operation on P(X). Witness of the fact that everything tends to be algebra, is the famus "algebraixation of mathematics".

You should also note that what you called " is closure under the defined binary operation", essentially it is included in the definition of an operation, binary and n-ary operations: @: XxX------> X. In general your question is a basic one that eevry mathematician should clear up.

I think regarding this issue could be interesting:

-Topological Groups, L. Pontrjagin.

-Topological Groups and Related Structures, Alexander Arhangel’skii, Mikhail

Tkachenko.

Dear Rey,

Topological groups etc. are rather a mixture of topological and algebraic structures. An algebraic structure is a structure where R=\emptyset. (In my notation above). The operation A-----> cl(A) is an operation on power set Boolean algebra. Thus Topological Boolean algebras are not regarded as a combination of algebra and topology. It might help to mention here the mother structure of Bourbaki: Ordered Sets, Algebraic Structures, and topological structures. All mathematics essentially is a combination of these three mother structures. See, e.g.

Leo Corry Modern Algebra and the Rise of Mathematical Structures- Second revised edition 2004.

The original idea of Bourbaki project was to find a foundation of Mathematics based on these structures, but: is there any demonstration that we can deduce all mathematical results from them?

And regarding original question, might be one can suspect that mixture of both topological and algebraic structures, shows that there exist essential differences.

Dear Rey,

Bourbaki have based their development on Set Theory ans Set Theoretic Structures. The only thing that it is not captured is Category Theory. For that Grothendieck invent "Grothendieck Universes". From a categorical point of view, Topological spaces and categories of algebraic strucutres are just categories. No difference. Also algebraic structures can be considered as relational strucutres, since an operation is just a functional relation. However there is a purpose to keep them separately. The intension is different. Formally, algebraic structure and topological one, they are both relational structures. The role in mathematics is different: Algebraic structures are for algebraic type of objects, whereas topological structures are for modeling: closeness, continuity and limits, i.e. analytic entities.

So, everything tends to be Algebra and we define other branches for applications? For example we defined Topology in order to work with the concept of point-wise and other kind of convergence & other functional properties? My perception is that although everything has -if you search deeper- to do with closeness, Mathematicians do not spend enough time to emphasize that fact. I remembered the professor of Topology who never answered my question":" Since we observe closure, isn't it based on Algebra?". Probably we have to do with the well played game : "this is mine, I repeat this is mine field and please stand one step away". But, with such a scientific egotism, Mathematics the only thing they have succeeded is to become a strange science with difficult notation and many hidden fights for the basic bricks... So, if you try to use maths in order to modelize universe for example, you will not be helped, unless you have the ability to look at the deeper essence of every math definition.

I agree with you Demetris! In all of my career, my only purpose was to find explanations for mathematical concepts, for myself and then teach them to students. This might be the reason that a student of mine in his Cambridge PhD, wrote:

"Finally, I wish to thank my teachers in Greece: My beloved teacher Kostas

Drossos, Dept. of Mathematics, Patras University, who introduced me to category

theory and gave me light to learn almost all of what I know."

Such statements is the return profit for teachers!

Topology is about nearness of sets and algebra is about variables known, unknown containg in an interval or region or set defined operations multiplication, addition,subtraction and division on variables. There is a relation between both algebra and topology called as algebraic topology in my research now i am able to define algebraic topology on near-fields over regular delta near-rings in N-group sapce This is the main difference between algebra and topology........N V Nagendram

In algebra union,intersection and complements of sets difference of sets can be described whereas in topology countable,uncountable,compactness,completeness and separatedness, connectedness of sets denseness,nowheredense and everywhere dense concepts can be described which is very interesting in topology.

What about motivating intuitions? Topology was developed basically to deal with intuitions about "space," "connectivity, "continuity," notions of "near" and "far," etc. Algebra came about in order to deal with notions of "finitary manipulation," especially in connection with equalities. The historical developments happened quite organically, without any particular "guiding hand," so the boundaries are fuzzy indeed...

I wonder if the boundaries are helping or not the Mathematical science to proceed. From my small experience I have seen many professors of Mathematics to act as a one dimensional scientific being, working in his/her field and not knowing about the neighbour one. Probably such a strong specialization has killed the intuition and prevents science to present new knowledge. I don't know...

As for the present question: If we define the terms of union & intersection for sets as n-ary operations, then Topology seems to be a 'subset 'of Algebra.

I agree, Demetris, that the Boolean operations make topology look like algebra, but you need to be taking unions over arbitrary families of sets, not just pairs.

From the point of view of modern model theory and universal algebra, there is a fixed list of relation and operation "names." This list is the similarity type, or the signature of the structures we have in mind, and is tied to a common vocabulary of relation and function symbols. From his point of view, a topological space cannot be taken as a set together with a list of unary relations (=open sets) because there is no fixed signature. The topology itself is a unary relation on the power set of the underlying set, and this fact makes an enormous difference. The lack of fixed signature tying topological spaces together may seem like a pesky little detail, but it's actually fundamental, for a host of technical reasons.

Among the structures that dwell in a "family of subsets", we have not only topology (in two versions: open sets and closed sets), but also rings (as defined in measure theory) and (abstract) convextities (first studied in the sixties/seventies). The latter are stable under arbitrary intersection and up-directed union. Although quite similar to topology, they are better suited for topics in convex and Euclidean geometry and its examples come largely from graph theory, universal algebra, ordered sets, and metric spaces. Filters (as used in topology) are another variation on the family-of-subsets theme. They found an important application in (logic) model theory through ultraproducts. Traditional Boolean algebras are yet another variation. With abstract Boolean algebras and their generalizations (lattices, semi-lattices) we definitely move into so-called Universal Algebra. Boolean algebra also leads to Boolean rings, which are rings as meant by traditional algebra.

Perhaps the only common theme of all these structures is "closure under one or more operations".

Almost like species in biology there is a fairly continuous spectrum of mathematical structures with close or far relationships. Unlike biology, we also have "combined species", like topological groups, topological convex structures, topological Boolean algebras, etc. Moreover, algebraic methods are applied in topology and in geometry. And real analysis is based on applying topological concepts (limit, connectedness, compactness) to the real field.

Dividing mathematics into "topology", "algebra", "analysis", "geometry" etc., is mainly done for clarity in (undergraduate) teaching. Mathematics should rather be seen as a highly coherent body of knowledge.

Paul has pointed to a technical difference between topological and algebraic structure. In the usual logic "jargon", an algebraic structure is first-order, whereas topological structure is second order. For this reason, topology depends more heavily on the use of set theory.

Paul is also right in pointing at the different intentions of algebra and topology. Compared with "intentional differences" the technical differences in two definitions are merely details. My favorite pair of examples is this: the closed-set axioms of topology are quite similar to the axioms of (abstract) convexity. You just have to replace "stable under finite unions" by "stable under up-directed unions". In both structures, there is a closure operator with fairly similar properties (it is called a hull operator in convexity). The property "cl(A) \cup cl(B) = cl(A \cup B)" of topological closure is weakened to "A \subseteq B implies cl(A) \subseteq cl(B)" for convex hull operators. In spite of a high similarity, the two theories have quite different intentions and behaviors. (It is like humans and primates having almost the same genes and a totally different life style.)

Honestly, I encountered few people claiming an in-depth affinity of topology with algebra. In contrast, a strong bond between convexity and universal algebra has existed almost from the beginnings of these two (fairly young) branches of mathematics. This bond is due to a seemingly small detail: a hull operator is "domain finite" in the sense that it is completely determined by knowing the hull of each finite set. Every reasonable definition of closure in any algebraic structure is domain finite. The bond is so strong that algebraic invariants may agree with (geometrically inspired) invariants of convex structures with a convexity defined from the algebra. Yet I am not aware of mathematicians claiming seriously that convexity is (universal) algebra. Again, the respective intentions are far too different for this.

It is amazing how many structures can arise about just from a simple demand: Closure. Are there any other so fundamental concepts, like Closure, in Mathematics? Because, as long as I am thinking, I have difficulties to find...

As mathematical disciplines, Algebra and Topology are different theories. But there are many interplay applications of Algebra to Topology and wise verse, such as Homology theory.

Demetris,

Being closed or simply Closure-ness is one of the fundamental properties we seek on any stable structure of either mathematical ( algebraic, analytic, geometric etc) or otherwise. The formal definition you indicated, given in text books of a topological space is in principle the algebraic requirements for any collection of pieces of the set that covers it. But in it, topological spaces are equipped with more properties and features than algebraic structures. Algebraic structures, which should also be closed under suitable binary or tertiary or unitary operations, are more involved in addressing solutions to algebraic equations, while topological structures are more suited in addressing problems of geometric nature - searching objects that are invariant under continuous deformations for instance, sub-domains that are connected or not, orientable or not, sub-domains which have boundaries, etc, which all are not doable by purely algebraic means. If we see further, meterizable topological spaces are vital not only for doing algebraic computations of a topological nature but analytic computations or simply analysis as well.

Yes indeed, dear friends. But my surprise is still enough and I have never answered the question about the existence or not of a second, such fundamental property like Closure-ness! Does it really exist, or everything are based on closure???

A professor told during a lesson I was attending : "Here is an algebraic description of a topological object" and after hearing that I did understand the very difference between them.

Mainly in algebra infinite repetition of operations is not defined whilst is is possible to give a meaning to in topology with the notion of limit.

More precisely, in a field for example you can add 2 or more elements, but if you try to sum an infinite number of elements it is not defined as an element of the field. Anyway, if the field is also can be equipped of a topology (this is the case of the field of real number), you can define the sum an a convergent infinite series, as the limit of the partial (finite) sums.

This is the same for vector space : in any linear combination there is a finite number of vectors involved. But this is possible, with topology (normed vector spaces) to define the sum of infinite weighted series of vectors.

@Demetris who love "closure operators"

One difference is, by algebra you may answer the topological questions but by topology you can not answer the algebraic questions.

One great insight that modern mathematics offers is that the space and the space of functions on the space are "dual" concepts in some sense. So the algebras and spaces

are somehow paired and can be thought of as being same. Here all these concepts have precise definitions but I cannot write due to length and complexity. Also, there are really no single way to see this duality. There are many subareas where this relationship are understood.

I do confirm that algebra can answer topological question (e.g. and invariant space associated to a differential equation is a finite dimension subspace of some vector space").

Conversely, topology do answer algebra question (e.g. "the sum o of an infinite real series is also a real number")

I think that using algebra we can translate topological proplems into algebraic one. The algebraic topology has as its subject exactly this. The other way around, does not possess a general method, only accidentally and single problems can be solved

A final remark is this: A topological space can be seen as a relational structure. Each open set can be seen as a 1-ary relation and so a topological space is a family of 1-ary relations. On the other hand an algebraic structure is just a set with a family of n-ary perations, n=1.2,...Thus formally both Algebraic structures and topological ones are just relational structures. Conceptually however there is a big difference, and for this reason we keep operations as "operations" and not as relations.

Costas, I agree with you that algebraic methods have traditionally helped topology more than the other way round. However, if you view topological spaces as unary structures, with each open set interpreting a unary predicate symbol in a first-order language, there is a problem because there is no cardinal bound on the number of symbols. If you impose such a bound--and hence a bound on the number of open sets you can have--other techniques of model theory (e.g., ultraproducts, Lowenhem-Skolem theorems) no longer apply. There are other ways to do topology from a model-theoretic viewpoint, but they usually involve lattices of open (or closed) sets that form bases for the topology. There is a fairly extensive literature on this topic.

Dear Paul,

I agree with you!

I think, in a structure (X,F,R,C) we do not impose cardinalities on the index of each family. Of course, what I stated that a topological space can be seem as a mathematical structure, does not solve the problem of developing a first order logic. This might be the reason that mathematicians prefer to study frames and locales.

I think that the difference comes from the fact that topology introduces the concept of proximity (that is close to but not necessarily identical to), which enables to extend any operation defined in an algebraic structure. The main case is the extension to an infinite number of arguments. A good example is vector space : they can be defined ina pure algebraic framework, and with the axiom of choice one can prove that any vector space admits a basis. But a vector can have only a finite number of non null components (even if there can be an infinite nuumberof vectors in the basis). The natural, topological extension, are Schauder bases, and furthermore Hilbert spaces.

In addition to the important points mentioned by @Costas, @jean claude, and @Paul Bankston, it possible to expand on this  interesting topic further and consider some  recent views of algebra and topology.

For example, one way to distinguish between algebra and topology is to consider localising a set A vs. localising a spectrum E.   This distinction is given in

I. Bobkova, Resolution of spheres, Northwestern University, 2013:

http://www.math.northwestern.edu/~bobkova/talkWIMS.pdf

For a much more detailed view of algebra vs. topology, see Section 2.2.3, starting on page 45, in

M.L. Guidice, My way to algebraic geometry.  Varieties and schemes from the point of view of a Ph.D. student, University of Bath, 2005:

http://www2.imperial.ac.uk/~jr311/calf/pdf/My_Way.pdf

Dear James,

 Good links, thank you!

Indeed algebra and topology are different and very very different. More specifically, algebra is precisely the place where the game of generators and relations can be played, and, ..... let me call it analysis...., is one which refuses to play the game in this way! It is fascinating that, seen in this way, there are many more examples of algebras than merely the finitary universal algebras ---- for instance the compact Hausdorff spaces shall be algebras over sets, and topologists knew this for a long time! In the study of compactifications, special importance used to be given to the Stone Cech compactification of discrete spaces, and them only! Reason: they are the free algebras, and every other compact Hausdorff space is a quotient of the Stone Cech compactification of a discrete space (i.e., the game of relations on the free algebra!)

Returning to this question, in particular. Much of topology can indeed be done using a familiar class of algebras known as frames. In fact, every statement in the theory of Hausdorff topological spaces can be reduced to a statement in the language of frames,  which can then be dealt upon using pure lattice theoretic arguments of frames. But, this is NOT the case for ALL topological spaces. Technically one could exactly reduce the Hausdorff-ness above to sobriety, where a topological space is said to be sober, if its irreducuble claosed sets are precisely the closures of singletons.

In conclusion: topological spaces is NOT algebraic, i.e., the game of topology cannot be accomplished with the generator-relation game plan!

From a point of view generated by mathematical logic and model theory, the difference between algebra and topology is basically a difference of order. Most algebraic statements can be written down using only first order predicate calculus. Topological statements are by their nature second order statements, between one has to quantify over all subsets of a set. As far I know, all tentatives to introduce topological algebraic structures in model theoretic frames were only partially succesful, in the sence that one could not express all topological facts, and had to introduce sintactic conditions for "correct" formulae. A very good example is the paper:

Model theoretic methods in the theory of topological fields.

Alexander Prestel; Martin Ziegler

Journal für die reine und angewandte Mathematik (1978)

Volume: 0299_0300, page 318-341

where "local" formulas are introduced, and then it is proven that they satisfy a restricted form of Löwenheim Skolem. The point is, that the studied topological fields have properties similar with fields whose topology is defined by metrics or valuations, which are already algebraic (first order) objects. This would not work for very General topologies.

The cited paper by Prestel and Ziegler can be found here:

https://eudml.org/doc/152007

Dear Mihai,

I agree with you. In previous post of mine I stated that: "I think, in a structure (X,F,R,C) we do not impose cardinalities on the index of each family. Of course, what I stated that a topological space can be seem as a mathematical structure, does not solve the problem of developing a first order logic. This might be the reason that mathematicians prefer to study frames and locales."

This I think clear up the situation. See also the post by Partha Pratim Ghosh.

Dear Mihal,

very interesting point of view. Analysis, by the notion of proximity, enables us to extend the scope of algebraic operations to operations comprising an infinite number of arguments. So it seems that analysis is clsely linked to infinity, how infinity is seen in predicates logic ?

@ Costas, thanks for the hint.

@ Jean Claude, there are some methods to approach infinity in predicate logic. One of them is non-atandard analysis. A later development, started by Edward Nelson, proposes a whole set-theory expanded with a unary predicate st(X) - X is standard. All infinite or infinitely small reals has to be non-standard... In the cited paper by Prestel and Ziegler, one quantifies over the neighborhoods of 0, so there is only an Approach for infinitely small elements, but this is enough. Generally, one cannot speak about points at infinity without introducing a filter of neiborhoods of those points, and this makes the structures almost compact. At this point we are again leaving the universe of first order logic. First order logic has models in all cardinalities (Löwenheim Skolem), but on the other hand compact sets are at most of cardinality of continuum, so a direct Approach is impossible. Only by sintactic tricks as done by Prestel and Ziegler with their local formulae, or by Nelson with his "standard" predicate permits us to approach the Notion of infinity from Analysis.

Yes, but it seems that the introduction of infinity is done through the set theory. I guess that it cannot be done properly inside the mathematical logic framework itself. Am I right ?

@ Jean Claude

Well, at the beginnings, one took only the ordered field of real numbers and did an ultrapower over a non-principal ultra-filter. With the canonical embedding of R in R*, one has already the standard elements. All other elements are non-standard, but " internal objects" - that means that one gets them over the ultrapower. There are some objects which are non-standard and external - they cannot be got by ultrapowers - as some external homomorphisms. Now, with this increasing menagery of objects, it became clear that one needs an axiomatic approach. As all objects can be constructed by set theory, it was Nelson's idea to put the standard predicate from the very beginning, and to create the IST = internal set theory. IST = ZFC + three new axiom schemes.

So, the answer is: no, one has many possibilities of doing non-standard analysis. One can start with real numbers and do ultrapowers (saturated models) and then embed the reals inside and work by hand (as Abraham Robinson did), or one can work axiomatically and look at the reals as they already contained non-standard elements, as Edward Nelson did.

If we think historically, the way Leibniz, Descartes, Newton, Bernoulli, etc... worked with small infinities when descovering calculus was much more in Nelson's style as in Robinson's style.

If we think historically, the way Leibnitz, Descartes, Newton, Bernoulli, etc... worked with small infinities when descovering calculus was much more in Nelson's style as in Robinson's style.

If I understand :

i) one of the most populat ZFC formulation has already an "axiom of infinity", which looks like the Nelson's function.

ii) it seems that the big issue is not to define countable infinity, or even continuum infinity, but going further.

Actually if one looks at most mathematial theories, they work well with countable infinities, less so with continuum (but they are a necessity for completion), and after that, they have little use. I think specifically to Hilbert spaces, which are quite manageable when they are separable, but not so with uncountable basis. So we could be lead to think that there is something which blocks our true understanding beyond coninuum infinity. Could it be somewhat formlised ? Saying that one could distinguish, in more than the usual cadinal theory, different kind of infinities ?

@ Jean Claude

Of course there is an Axiom of Infinity in ZFC, but this should not be confounded with E. Nelson's Extension of ZFC for nonstandard Analysis. The Axiom of infinity says that there is a set I containing the empty set, and for every set x in I, x U {x} is in I. This permits only the definition of natural numbers like 0, {0}, {0, {0}}, {0, {0}, {0, {0}}}, etc.

The infinity we were speaking about since now was only the symbol $\infty$ used in Analysis to define or to express several limits. It was not a question of cardinality since now.

I prefer to sum up the discussion since now, instead of making it too wide:

- ZFC has a language containing only one relation, "in", used like "A in B" (A is element of B)

- ZFC has an Axiom of Infinity that assures the existence of smallest infinite sets.

- ZFC proves the existence of an infinite number of cardinalities (infinite sets with different cardinalities, because there is no bijection between them). In fact, for every set A, the set of subsets P(A) is strictly bigger than A.

- ZFC proves that a first order theory that accept an infinite model, will accept models in all bigger cardinalities. (Löwenheim Skolem upwards)

- In ZFC one can construct the real numbers, introduce the Symbol $\infty$, and make the whole Analysis.

- In ZFC one makes ultrapowers. An ultrapower of the field of real numbers contains infinite and infinitely small elements. Infinite elements are bigger than every integer, infinite small elements are strictly positive and smaller than every positive rational. With their help one can make non-standard analysis like Leibniz, Newton, etc...

- There is an Expansion of ZFC called IST, did by Edward Nelson, in a language consisting of the relation "in" and a new predicate "Standard(x)". In this language and using three new Axiom schemes, one constructs directly a model of non-Standard Analysis, without needing ultrapowers.

- Different from Algebra, topology does not have models in every cardinality. The example was the compact spaces, which can be only finite, countable or of cardinality of continuum. (I don't know what happens if one accepts also cardinalities between countable and continuum, maybe there are still some intermediary compacts...) This was my difference between algebra and topology. Topology is NOT first order from the point of view of logic.

- From the point of view of non-Standard Analysis, something is compact if and only if every point is near-standard (infinitely close to a standard point).

OK. Now to your question. It seems that you say that we cannot well understand objects of cardinality bigger than continuum from an analytic point of view. (we underszand them from an algebraic point of view, because for the algebraic theory we always have a countable model, see Löwenheim Skolem). I guess that the reason is that our understanding is too Close to some Special topologies, like thos which are locally compact, and that exactly those properties are missing in bigger cardinalities.

If I don't understand well the question, please put it in a closer relation with the summary of the discussion, as tried in the lines above.

OK,

I Ithink that I understand better, thanks you. To avoid misunderstanding I think that we should precise the kind of topology which is involved.

What it seems is that one can build topology (meaning sets with the basic axioms of a topological space)  with a standard ZFC, it can include the integers and cardinality theory. But, for instance to build real numbers, one proceeds by completion, and completion requires a metric, and if one looks at the definition of a Cauchy sequence it is clear that somewhere "small numbers" should appear.This can be done either by what you call ultrapowers, or by extending the ZFC model.

When you say "topology does not have models in every cardinality" it should be precised : some topologies (such as topologies requiring not localy compactness) can exist only with sets of cardinality greater than the continuum.

I don't see precisely where the predicates appear : it seems that some of these constructions require more than first order predicates.

Dear Mihai,

   I do not quite understand the meaning of the statement:

The example was the compact spaces, which can be only finite, countable or of cardinality of continuum

   in your answer, for there must be compact spaces which have larger cardinalities than the continuum. I am taking the meaning of cardinality of continuum to be what is usually denote by c, which equal to 2^{\omega_0} under CH.

However, lets take a set X of the cardinality of continuum, and consider its power set P(X). Definitely, P(X) has a larger cardinality than the cardinality of the continuum, and consider P(X) as a discrete topological space. let BP(X) be the Stone Cech cpmpactification of P(X), and we know that P(X) is denseyl embedded in BP(X), and is obviously not isomorphisc with BP(X). Hence BP(X) is a compact Hausdorff topological space with cardinality larger than the continuum.

Taking subspaces one then also obtains examples of topological spaces which are of every cardinality smaller than BP(X).

With my regards,

@ Jean Claude

For constructing the reals, one can use the Dedekind cut method (cuts in the rationals; a cut is a partition of the rationals in two convex sets) - and that does not require a metric a priori.

The statement  "topology does not have models in every cardinality" meant something like topological properties, as compactness, which are not possible in all cardinalities.

Predicates appear in the following way: if one takes the first order theory of the field R with symbols +, -, . , < (the set of all formal Statements, with quantifiers, true there, using these symbols) one has a theory (called theory of real closed fields) which has models in all cardinalities. But if we ask that the order should be complete [that can be done only by an Axiom of second order - it means, one quantifies over all subsets], one gets an axiomatic System with ONLY ONE model. This model will be the field of the reals.

This is the difference between first order and second order theories. First order theories, if they have infinite models, then they have infinite models in all cardinalities. Second order theories can have models only in some infinite cardinalities, or only in one cardinality.

The compact sets can be only finite, countable infinite and continuum. So topology is essentially of second order.

Algebra is essentially of first order, because algebraic axioms refers normally only to elements, and not to subsets.

To your last question: there is a definition of the reals, done inside ZFC, which uses only the language of ZFC [that is, the symbol $\in$] and so is first order (!). This is OK, because we expect that every model of ZFC contains a field of real numbers. And now comes the paradox: there are countable models of ZFC (!) because ZFC is a first order theory. Looked from outside, the field of the reals living there is countable. But internally, it is uncountable, and there is no internal bijection to N (internally refers to this special model of ZFC).

So, as you see, thinks are very subtle and complicated....

Thanks for the nice discussion!

@ Partha Pratim Ghosh ·

RIGHT!!!! I meant all the time compact metric spaces. They are separable (contain a countable dense subset), so cannot be bigger than continuum. But the rest of Argumentation remains true.

Algebra and topology are completely different branches of mathematics. Yes the word "closure" appears in both but as different concept. In algebra we talk about the closure of a binary operation; in topology any subset admist a closure (smalest closed set containing it). Algebra is essentialy discrete but topology studies continuity ( the opposite of discrete !) Algebraic topology is the use of algebraic structures to study and classify topological spaces (such as surfaces). 

The following book may be of interest to followers of this thread:

Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, 2011:

http://www.maths.ed.ac.uk/~aar/papers/barmak2.pdf

Dear Jacques,  I think the word "closure" has the same spirit in both algebra and topology.  On the algebraic side you can talk of "subset A being closed under the operations of X;" on the topological side you can have "subset A being closed under net convergence in X."  On the one hand you have subalgebras, and on the other you have closed subsets.  The "closure" theme has many manifestations, but basically the same motivation throughout.  (Mostly, anyway.)

Demetris writes "why have we defined two branches that are almost of the same philosophy?. For me this is  a vague philosophical statement that has no mathematical sense. Algebra studies operations +. . , *, U, etc. Topology studies spaces, continuity, surfaces, curves. Look at Algebra (by Van der Waerden) and Topology (by Munkres)

These two books have ALMOST NOTING IN COMMON.

Read in MathWorld.

I refer back to my Jun 20 posting.  I think what is muddying the distinction between algebra and geometry/topology is the advent of the set-theoretic foundations  of mathematics.  Since Cantor-influenced Bourbaki, everything's a set!  But if you go back to pre-Bourbaki days, mathematical objects are quite differently presented:  groups comprise "invertible operations;" geometries comprise "lines" and "points."  Points are not viewed as elements of a line, but rather are related to lines by the notion of "incidence."  Motivation's the thing.  Somebody else may come up with a proof that "you're both studying the same thing,"  but initially you have your own focus and impetus.  An example is Stone's duality theorem, relating Boolean algebras and totally disconnected compact Hausforff spaces.  Here's an overarching--and formal--connection between the purely algebraic and the purely topological.

@ James: nice book! Thank you!

@ Paul & Jacques: There are people doing  branche-related (domain-related) mathematics, and other people doing  problem-related matehematics. The first class say "this is the theory I love, so I work inside". The second class say "I would like to solve this problem, doesn't matter what theoretic facts I apply". I have big respect for the first class, but I feel more sympathy for the second. Well, sentiments don't matter at all, of course. However, people from the first class tend to see more the differences; and for the people in the scond class the whole mathematics is a continuous field with a fluid landscape, where all is permanently mixed. But both parties are right!!!

As to the notion of closure, I tend to be at the Paul's side. The topological closure of a set is the intersection of all closed sets containing the given set. The subgroup generated by a set is the intersection of all subgroups containing the set - and the same for subrings, subfields, affine subsets, etc. The closure operator is in both cases monotone, expansive (means A subset cl(A) ) and idempotent (means cl(cl(A)) = cl(A) ).

Algebra and Topology are totally different branches of Mathematics. In Algebra the the closeness is as necessary property of the Set. The set must be closed under said binary operation. In Topology the closure is used on a set or its subset. We say that closure of A is the intersection of all the closed sets containing A. In Topology we talk about openness,  closeness, continuity,  homomorphisms, connectedness,  compactness,  Housedorffness etc. These properties do not occur in Algebra. Also we use a lot of analysis in Topology.  I think these are enough fir two different names.

I agree 100% with Ellis D. Cooper: Saunders Mac Lane book, "Mathematics Form and Function" is a must. (with parts on Mechanics). "Mathematics and Its History", by John Stilwell is also one of my favorite. Geometry, Algebra, Analysis, Number theory, Probabilities, Topology are all different branches of mathematics, and all very relevent. To say that one is useless, as part "philosophicaly" of another, is WRONG. Long time ago Bourbaki wrote: Probability is a part of mesure theory were the whole space has mesure 1. As such it is irelevent ... Ask actuaries, probability and statistics departments what they think of that ....

Dear Ellis,

I know McLane's book. I think he has a big network of branches of mathematics, whivh I find not so confortable. I prefer Lawvere;s books and philosophy. The books 

LawvereS-Schanuel_Conceptual_Mathematics

Lawvere-Rosebrugh_Sets_for_Mathermatics

The philosophy of Lawvere is concentrated on two options: Set theory or Topos theory,

which might include McLane's network, but at the same time focuses on the important issues. 

We agree. McLane's book is nice, but it is a survey of various fields of mathematics. As such it is valuable since it is McLane's. It is good as an introduction for non mathematics majors. On the other hand the Lawvere and other books covers in an elementary fashion Categories and toposes, which constitute a kind of holistic foundation for  mathematics. They are more useful for foundations of mathematics persons. 

For me praktical topology are relationship (neighborhood)

Hi, the major difference between Top and Alg is, that the latter doesnt have initial structures, eg not every subset is subobject, eg there is no product of fields. More exact, the forgetful functor is topologic for the former and algebraic for the latter. However,  Top intersected Alg is Comp.

Sorry, I didnt had time to read all your posts, Joachim

Similar to that between two strands of a double helix! Functional Analysts for ever trying to create fusions of the two!

In addition to what @Paul Bankston has observed, a basic distinction between algebra and topology stems from the contrasting views of sets.    For topology, the focus is on the nearness of points to sets.   By contrast, for algebra, the  focus is on the filtering of sets that is made possible by defining binary operations on sets and on the introduction and study of homomorphisms between algebraic structures such as groupoids.   

usually algebra concerning with X-axis or on simple XY-plane but topology is with some space on XY-plane.

Iniltially Demetris wrote: What is the essential difference between Algebra & Topology?

Both of them are guided by the concept of closure. But the term closure has completely different meanings in both branches. The two branches study different objects, have different methods, have different problems, ... What's the difference between physics and Chemistry: none, they both study mater. Why Calculus and Analysis: they both study differentiation and integration. 

Dear Jacques,

using the same reasoning as yours, we may say that all sciences do not have any difference: They all study nature or matter! But they are really differences if you detail your objects.

Dear Costa,

that's exactly my point and that is why algebra and topologie are two different branches of mathematics. As you write: They all study, here sets and subsets but they are really differences if you detail your objects.

OK, we agree dear Jacques!

Except for cross-word and sudoku puzzles, newspapers and magazines sometimes offer amusement by printing two seemingly identical drawings, with a few minor changes in the second. The title is (usually) "find the seven differences". A magazine once made fun by presenting two totally different pictures under the title "find the 7923 differences".

I understand Jacques' post as irony of this kind. Topology and algebra are hugely different.

Yet they also cooperate quite well (in topological algebra and in algebraic topology), and have an important concept of closure in common. I find this thread valuable for mentioning several "meeting places" like pointless topology in Heyting algebras, topological fields with a topology derived from a valuation, and Stone duality between topological spaces and locales. Please go on.

Phrased in terms of puzzles, we have two totally different pictures with title "find the seven similarities".

I do not think that this is a well-posed scientific question. Newertheless, one may observe that algebra lives in the left hemisphere of a mathemcian's brain, while topology in the right one. So, why the evolution created these hemispheres as an universal instrument of understanding, i.e.,  of abstract modeling of what we are interested to domesticate, that is the question that worths a discussion. A mathematician can contribute to the solution of this fundamental problem by analyzing how Algebra and Geometry interact in solving math. problems, The history of the Fermat problem is very instructive in this sense.

Dear A. M. Vinogradov,

I recognise your nice opinions in

R.V. Gamkrelidze, E. Primrose, D.V. Alekseevskij, V.V. Lychagin, A.M. Vinogradov Geometry I. Basic ideas and concepts 1991. 

I could not agree more with your opinion!

Better you may think about the beautiful connection and interaction between Algebra and Topology. Of course it is more rewarding

Pierre Basieux discribes in his book "Die Architektur der Mathematik" (I just found it in german language) there are three fundamental types of mathematical structures: (1) order structures, (2) algebraic structures and (3) topological structures. On page 68 there is nice illustration i'd like to explain.

Let M be a set. Now look M as structured respectively by

So, for me this is the difference your asking for.

Dear Detlef,

These structures are the "mother structures" of Bourbaki!

Yes, of course. It's mentioned in that book, too, but later on by discribing mathematics as a building out of these structures. Maybe the author told it also before, so your right, I also had to mention Bourbaki. I thought about it and discarded this. The point for me was, that Topology and Algebra investigate different structures.

My note was made more of interest by my side. Topology up to now was just a marginal topic for me. I've seen later, that this question is discussed over 7 pages and I'm sure I can't contribute something more here. For me, it seemed to be the point of the most other answers i read.

Oh, I found your answer on the first page, Costas. Sorry for my repeating.

Algebra is basically knowing about or studying about elements of sets which are empty or non-empty that is union intersection and complements addition subtraction quotient sets what not and its existence in space whereas topology is studying about nearness of sets that is closure disclosure dense everywhere dense nowhere dense and  connectedness disconnectedness compactness finally completeness of all set which are described thereof in algebra and many more differences are there likely to discuss a platform you can call seminar / conferences which is useful to meet where the mathematicians required place to discuss important things and issues about difference of algebra and topology it is not one word difference as we feel instantly in my opinion what do you say but you can satisfy with the answer i am given.....N V Nagendram

Algebra is the geometry of notation and topology is an abstract geometry. Geometry could be said the algebra of gemetric objects. Indeed there is a discipline of matematics called algebraic geometry and another one called algebraic topology

The question is rather deep, although phrased bit weirdly! A short answer comes from Beck's Monadicity Theorem, wherein an algebra is exactly described as precisely those places wherein one can make sense of generators and relations. Topology, obviously is not of this kind, and there are many others obviously which are not of this kind. For instance, locales in alike topology, i.e., is not algebraic, but frames (the dual category of locales) is algebraic!

Algebra live in the left hemisphere whereas topology live in the right. Since the two hemispheres are connected with corpus callosum one expects a similar synthesis, so that we have Algebraic Topology, algebraic geometry etc.

The boundary between algebra and non-algebra is actually quite fuzzy. For instance, latin squares are square matrices of size n, filled with n symbols (usually: 0,1,..,n-1) each appearing once in each row and in each column. On the one hand, a latin square can be seen as the Cayley table (multiplication table) of a so-called quasi-group, where products with an unknown have a unique solution. That's algebra.

On the other hand, latin squares can be clustered into MOLS (Mutually Orthogonal Latin Squares).  The existence of a MOLS of size n-1 is equivalent with the existence of an affine plane of order n (see Brualdi and Ryser, theorem 8.4.12), which in turn is equivalent with the existence of a projective plane of order n. That's geometry.

Mathematics does not quite respect the left-right division of the human brain.

N.B.: Brualdi-Ryser's book is published by Cambridge University Press 1991, in the series "Encyclopedia of Mathematics and its Applications", Vol. 39.

Article Combinatorial matrix theory / Richard A. Brualdi

Dear Marcel,

You choose a very special case in order to describe the difference between algebra and topology. furthermore you made the statement that “Mathematics does not quite follows the left-right division of the human brain.”

First I quote from Kostrikin-Shafarevich, Algebra I, § 1. What is Algebra?: “One can attempt a description of the place occupied by algebra in mathematics by drawing attention to the process for which Hermann Weyl coined the unpronounceable word 'coordinatisation' (see [H. Weyl109 (1939), Chap. I, §4]).”

 So Algebra essentially is “cocordinatisation”.  As for the lateralisation of the human brain and its relationship with mathematics I have to say: There are some people that refuse the lateralisation of the brain. Even so we should “invent” such dichotomy since it has a very strong explanation of the substance of mathematics. Especially the Russian geometers pay very much attention to it. Let me quote some of them:

• Manin, Mathematics and Physics, Birkhouser, “On a more special plane, it has become possible in recent years to consider the relation between mathematical symbolism, informal thought and the  perception of nature from a point of view provided by new data about the structure and functions of the central nervaus system.

 The brain consists of two hemispheres, a left hemisphere and a right one, which are cross-connected with the right and left halves of the body. The neuron connections between the two hemispheres pass through the corpus callosum and commissures. In neurosurgical practice there is a method of treatment of severe epileptic seizures, in particular, which consists of severing the corpus callosum and the commissures, thereby breaking the direct connections between the hemispheres.

After such an operation patients are observed to have "two perceptions". In the laconic formulation of the American neuropsychologist K. Pribram, the results of studies of such patients, and also patients with various injuries of the left and right hemispheres, can be summarized in the following way: "In right-handed persans the left hemisphere processes information much as does the digital computer, while the right hemisphere functions more according to the principles of optical, holographic processing systems." In particular, the left hemisphere contains genetically predetermined mechanisms for understanding natural language and, more generally, syrnbolism, logic, the Latin "ratio"; the right hemisphere controls forms, gestalt-perception, intuition.

Norrnally, the functioning of human consciousness continually displays a cornbination of these two components, one of which may be manifested more noticeably than the other. The discovery of their physiological basis sheds light on the nature and typology of mathematical intellects and even schools working on the foundations of mathematics.

•  In D. V. Alekseevskij, A. M. Vinogradov, and V. V. Lychagin, GEOMERTY I: Basic Ideas and Concepts of Differential Geometry Springer 1991., we find: “§ 1. Algebra and Geometry -the Duality of the Intellect

 It is known from physiology that in the process of thinking the hemispheres

of the human brain fulfil different functions. The left one is the site of the

"rational" mind. In other words, this part of the brain carries out formal

deductions, reasons logically, and so on. On the other hand, imagination, intuition,

emotions and other components of the "irrational" mind are the product

of the right hemisphere. This division of labour can have the following

explanation.

The process of solving some problem or other by a human being or an artificial

mechanism involves the need to draw correct conclusions from correct premises.

The logical computations that carry out these functions in various specific

circumstances can easily be formulated algorithmically and thereby carried out

on modern computers. There are good reasons for supposing that the human

brain acts in a similar way and the left hemisphere is its "logical block". However,

the ability to argue logically is only half the problem, and apparently the simpler

half. In fact, to solve any complicated problem it is necessary to construct a

rather long chain, consisting of logically correct elements of the type "premise conclusion".

However, from given premises it is possible to draw very many

correct conclusions. Therefore it is practically impossible to find the solution of

a complicated problem by randomly building up logically correct chains of the

form mentioned, in view of the large number of variants that arise. Thus the

problem we are posing is: in which direction should we reason? We can solve it

only if there are various mechanisms of selection and motivation, that is, mechanisms

that induce the thinking apparatus to consider only expedient versions.

Man solves this problem by using intuition and imagination. Thus, we can think

that the process of evolution of nature has led to the two most important aspects

of any thought process - the formally logical and the motivational - being

provided by the two functional blocks of the brain- its left and right hemispheres,

respectively.”

In the book Jet Nestruev, Smooth Manifolds and Observables. Springer GTM, we find in the appendix:

A. M. Vinogradov Observability Principle, Set Theory and the “Foundations of Mathematics” we find: "The construction of such an overall picture, in other words, of the geometric image of the problem, takes place in the right hemisphere, which was created by nature precisely for such constructions. The basic building blocks

for them, at least when we are dealing with mathematics, are sets. These

are sets in the naive sense, since they live in the right hemisphere. Hence

any attempt to formalize them, moving them from the right hemisphere to

the left one, is just an outrage against nature. So let us leave set theory in

the right hemisphere in its naive form, thanks to which it is has been so

useful."

Finally I suggest to accompany your studies with some dose of foundations and philosophy of mathematics. F. Zalamea Synthetic philosophy of Contemporary mathematics, is a bomb book especially for you! 

Dear Costas,

I downloaded Zalamea's work and started reading. From the past I remember some disagreements between us, though my impression was (and still is) that our opinions are not that different. Rather, my way of expressing things is a bit one-handed.

Back to the current discussion: I need not be convinced of the different functionality of the left- and right part of the human brain, nor must I be convinced that various parts of mathematics may involve different parts of the brain.

I just object to nice (but too simple) characterizations like "Algebra essentially is coordinatization” (which is rather  a two-way bridge between algebraic manipulation of formulas and geometric manipulation of patterns). When I said that "mathematics does not quite respect the left-right division of the human brain" I refered to the example of latin squares to illustrate that ONE concept can link up with BOTH parts.  It would be an interesting experiment to scan a mathematician's brain when she/he thinks alternatingly of a latin square as a combinatorial pattern (a formal square equipped with some kind of state ) and as an algebraic operation that multiplies objects and obeys certain rules.

There are, in fact, lots of examples. For instance, lattices are both an ordered structure and an algebraic object with two binary operators (sup and inf). One of the most remarkable examples (in my opinion) are the median algebras (a median is a ternary operator obeying axioms generalizing the familiar median of numbers). These universal algebras are equivalent with (abstract) convex structures having the Kakutani separation property and having Helly number 2. If you know the meaning of these terms in standard Euclidean space and that these definitions are simply re-interpreted into "abstract" convex sets, you may appreciate my use of the qualification "remarkable".

This is the way mathematics stands above our brain partition.

Dear Marcel,

You should notice that the two hemispheres are connected with the "corpus calosum" which plays a role similar with adjunction of the two worlds. The examples you mention together with "Algebraic Topology", "algebraic geometry", etc. are just examples of synthetic mathematics that combine in a dialectic way the two types of mathematics. Adjunctions according to Lawvere represents in a mathematical way "dialectics". Zalamea is full of such dialectic works, describing Grothendieck's work. I hope that after Zalamea (a difficult work!) we should discuss again, you beem not one handed but two headed!

Algebraic topology is a theory, a huge conglomerate of cooperating algebraic and topological objects and methods. It is like a company with a labor department operating machines and moving things around on the one hand, and with an intellectual department doing the planning and the bookkeeping on the other hand. That's somewhat like the brain with its hemispheres cooperating for powerful results.

Objects like latin squares, lattices, median algebras, are almost atomic objects of mathematics with two fully equivalent faces: one algebraic, one geometric. They are not a cooperation of two separate parts, they are two-in-one, each constituting the whole.

The theory of median algebras is remarkably ambiguous: depending on which face is made dominant, quite disparate results appear, the other face becoming a mere commodity. The third reference is highly geometric, the first and  second are  involved with generating the algebra and are rather computational. Median algebras (from the early 1950's) and the corresponding convex  structures (from 1970's) have existed separately for about two decades before it was discovered that they were the same object.

Article Determination of msd(Ln)

Article The Median Stabilization Degree of a Median Algebra

Article Embedding Topological Median Algebras in Products of Dendrons

The word `epistemology' can sound as an alien in the discussion. Perhaps it will make sense.

Considering an epistemological perspective, one could say that algebra and topology belong to very different categories: the first is a fundamental domain; the second is an integrative proceeding between different domains. One consists of elements and the other is made of `adaptative' rules. This would agree with the position of some participants.

Different from logic, the epistemological view should give some explanations -outside mathematics, for this difference of levels, clarifying why algebra is in such a fundamental position. The explanation is that `algebra' is the domain of matrices and, originally (in epistemological sense), matrices are not related to numbers and to continuous spaces and/or geometry either.

The word origin can be used in several senses such as logical, historical, psychological, cognitive, physical, chemical, epistemological, and so on. In all senses, it involves the image of a center with detachment from it, where the description of this `detachment' will give the meaning for the `origin'. A historical detachment consists of series of facts after a starting point; the psychological consists of interrelated psychological facts; the chemical detachment would consist of growing structures; the physical can be made of spatial distances; the mathematical can be made of derivations, etc etc. It is just an example of a `pattern of thinking' behind different sciences.

Considering the usual history of ideas, matrices originated from systems of equation. As a mathematical concept, this history may be OK. But in an epistemological sense, one can argue that it was the other way around, equations were naturally systematized because  `tendency to systematize' was present already. In fact, similar patterns of systematization were present in other cultural manifestations, since early times.

The point is that this model of systematization (giving rise to matrices) and the model of centrality (the alchemical model) seem to be different patterns of thinking. This can also be an argument for this `fundamental' position of the algebra, bringing its own objects to mathematics. This is how I can see the situation.

Dear Claudia,

I did not get it! Mathematics (Algebra, Topology. etc.) have an epistemological problem. One must explain in what manner mathematics contribute to knowledge. I disagree that "the first is a fundamental domain; the second is an integrative proceeding between different domains". We know that mathematics undergo many reformations, one of them was the arithmetization and then algebraization of mathematics. This does not give precedence to algebra over topology. Then you say "One consists of elements and the other is made of `adaptative' rules.". Really I do not understand why algebra consists of elements and not operations on elements. Topology, at least poit-set topology consists of elements. "Adaptive rules"? Do you mean axioms? I think you should explain to us terms like:

What do you mean by "epistemology"?

What do you mean by "adaptive rules"?

There are infinitely many topological spaces and algebraic structures. Thus, in this question both terms algebra and topology are generic and the differences must be handled from a categorical view point. See the differences.

Every topological category satisfies the following properties.

1) Is initialy complete.

2) Is finaly complete.

3) Is fibre complete.

4) Has discrete structures

5) Has indiscrete structures.

By contrast, the forgetful functor F of every algebraic category satisfies the properties below.

1) F reflects isomorphims.

2) F reflects limits.

3) F reflects equalizers.

4) F is faithful and reflects extremal epimorphisms.

5) Mono-sources are F-initial.

@Claudia: In addition to Costas's and Juan's remarks, I would add having a problem with the statement that algebra is the domain of matrices.

If algebraic objects have to be described directly as a domain of knowledge, I would think in the first place of a domain called "universal algebra": one or more operators each with zero or more arguments operating on a set (or a combination of a few sets). This usually goes with some axioms describing "computation rules". Are these "adaptive rules"?

@ Costas: "One must explain in what manner mathematics contribute to knowledge"

That question could be asked of any human activity. Why in particular for mathematics?

Dear all, many thanks for the reactions.

The clarification of this mathematical question is very important for physics. I am a physicist and see it in the context of the development of theoretical physics.

Epistemology is the science which studies the development of other sciences. One of the interests is to establish connections between the different sciences. Theories are another concept, not all sciences are theoretical in the sense of having a structured body of knowledge, according to some rules. Rudolf Carnap gave important contribution on this topic but apparently made no impression on physicists. Physics develops, not only without structure but even rejecting it, what provokes a lot of `intellectual pain', specially on young people.

This question, about the difference between topology and algebra, reflects to me a similar situation, a call for some kind order in the science. Anyhow, I would not like to disturb the line of thinking, centered on mathematics, and I am not sure if the discussion on epistemology would be of the interest of this community. If yes, I can continue, as far as I can, trying to answer the questions which were posed to me. Otherwise one can start a different question to fit a discussion on the development of mathematics entangled with physics. Perhaps, in this way, one question can help the other.

Dear Demetris,

you have written:

'What is the essential difference between Algebra & Topology?

Both of them are guided by the concept of closure.

So, why have we defined two branches that are almost of the same philosophy?'

The natural answer to your question is that Algebra and Topology are two different structures of Mathematics, hence they follow the same rules !

Of course these different structures can concur to study the same object. In fact Algebraic Topology is just the great unification of Algebra and Topology.

Let me add also that even if from the formal point of view there is not methodological difference between Algebra and Topology, there is a great difference when they are applied in Mathematical Physics. In fact algebraic properties are often seen therein as local properties, when instead topological characterizations concern global properties. Then in this context Algebraic Topology allows to unify local and global properties ...

By conclusion, the difference between Algebra and Topology does not concern their methodology, but uniquely the way in the which they are used.

My best regards,

Agostino

The essential difference is: Tology is an order structure whereas Algebra is not an order structure.

It is a bit pointless to ask for the "essential" difference. Both algebra and topology are among the standard toolboxes of working mathematicians. It is very fruitful to be able to change perspective, and to go back and forth, often in non trivial ways, and to use constructions and theorems from different fields. 

Large swaths of topology can be reinterpreted as algebra.  This is because for "conventional" topological spaces (i.e. locally compact Haussdorf) the space can be recovered from the C^* algebra of continuous function as the space of maximal ideals or what is the same, a homomorphisms to the field of complex numbers (because by a theorem of Naimark, the only field that is a complex Banach algebra is C). The so called Gelfand-Naimark correspondence is very simple: every point x of the space X corresponds to a homomorphism C_0(X) --> C,  f --> f(x) where C_0(X) is the space of continuous functions that become arbitrarily small away from a compactum K \subset X. The Gelfand Naimark theorem says that those are the only homomorphisms and conversely we can interpret every Abelian C^* as a space of functions on the space of homomorphisms to C by evaluation[1].The idea of Noncommutative geometry by Connes and others is to consider general C^* algebras as "functions" on "topological spaces", and consider irreducible * representations as some sort of "point".

There is at least one completely algebraic version of this invented by Grothendieck: the 1-1 correspondence between the spectrum of a commutative ring spec(R) and the ring itself. It comes with a purely algebraically defined topology, the Zariski topology, which is a bit weird (i.e. needs getting used to) because it does not separate points.  It is a far reaching  (and, once you get the hang of it, very natural) generalisation  of the correspondence between the geometry of the solution set of polynomial equations

F_1(X_1, ..., X_N) = 0

...

F_M(X_1, ...,X_N) = 0

and the algebraic object of the polynomial ring (say over a field) modded out by the ideal generated by the polynomials i..e.

k[X_1,...,X_N]/(F_1, ...., F_N).

It is an extremely useful point of view, even if you are only interested in commutative algebra because it allows to make rigorous sense of the idea that localisation at a prime (in the algebraic sense) is localisation in the geometric sense. It also allows to do geometric things like resolving singularities, which go beyond spectra of a ring and therefore commutative algebra. It is fair to say that in algebraic geometry commutative algebra, geometry and topology (e.g. in the notion of properness)   have become completely mixed.

Conversely algebraic topology has had a tremendous influence on algebra. A key example is homological algebra.  Homological algebra is pervasive is modern algebra and is directly rooted in homology theory and Stokes theorem for differential forms, the proper (and more user friendly) generalisation of the Stokes/Gauss theorem. It is also completely fundamental in algebraic geometry but it would go too far to go into detail here. 

Algebraic topology actually often feels much more like algebra than like geometry. More generally, it turns out that a great number of purely algebraic problems can be fruitfully encoded in nicely behaved topological spaces and considering them up to homotopy[2]. 

For example consider a group G, about as clearly algebra as it gets. There is a topological space BG with G as its fundamental group and contractible universal cover EG. It is unique up to homotopy, but the usual models are so called CW complexes, topological spaces that you can think of as spaces built up from finite dimensional simplices. It determines the group G, because we can recover G as its fundamental group, but some things are more natural to see from BG then they are from G itself.   E.g. its group cohomology in a representation V is a purely algebraic invariant[3]. However, it is also in a natural way the topological cohomology with values in the local system EG\times_G V. The geometric point of view gives approaches to compute this group (the Atiyah Hirzebruch spectral sequence), and conversely shows how group cohomology comes into play in computing topological invariants of topological spaces. If one wants to, this cohomology can even, (at least for the trivial representation k), in turn be computed as the homotopy classes [BG, K(k,n)] where K(k, n) is the Eilenberg MacLane space, itself a generalisation of BG = K(G, 0). Therefore the cohomology theory is in some sense represented by the sequence of spaces K(k, *). 

There are much more advanced examples (e.g. algebraic K-theory) for which the general philosophy is that working with spaces up to homotopy (and possibly stabilised by suspension) gives a great deal of extra freedom, and conversely when axiomatized (e.g. in so called model categories)  topological constructions give nice and streamlined approaches to purely algebraic theories often involving some sort of algebraic notion of homotopy equivalence.   

Essentially all mathematics at some point involves some sort of algebraic manipulation, much of mathematics is influenced by geometric ideas.

[1] The space of maximal ideals inherits a natural topology. In fact the continuous functions also determine the continuous functions between locally compact topological spaces: they can also be recovered as the *- homomorphisms  between the spaces of functions. This gives a contravariant equivalence of categories between Abelian C^*-algebras and locally compact Haussdorf spaces. 

[2] Technically such spaces are CW-complexes, and up to homotopy they can alternatively be considered as simplical sets (which is higher dimensional generalisation of a graph) up to homotopy (which can be defined algebraically). Thinking of them as spaces greatly guides intuition and often proofs.  E.g. BG is a quite concrete space: the quotient of a nice contractible space by  that one can construct explicitly (the nerve of the category consisting of a single point * and Mor(*,*) = G), modded out by the group G. For some spaces however there are direct constructions. E.g. BZ = S^1.

[3] It is the higher derived functor of the functor of taking G-invariants, i.e.

Ext^i_{kG}(k ,--).

Dear Demetris, I will try to show the basis of my epistemological approach to the question you posed.

For the sake of argument, let us drown our minds in a different period, around 20.000 years ago or so. Then, we could say:

1- We can compare and classify shapes, admire and even adore them, feel good or bad with their harmony or disharmony. We know how to reproduce them on the ground, inside caves, or somewhere else. We do this for some practical reasons, communication or other purposes.

2- We also think of numbers. We make signals for them and use them for many purposes, including communication.

3- ....Still unknown.....

4- So, one day, we started to apply numbers to lengths and vice-versa, resulting in measures of shapes and in new numbers.

After this experience, we are back to here and now.

5-Comments and questions.

5.1- In fact, why did we connect shapes with numbers?

5.2- These domains of thinking (1,2) are not the same, they do not even need each other to exist and could have had separated existences for ever. There are registers that a period (1,2) happened in history and still happens as psychological stage. We can also observe our modern human babies and children, showing some understanding of these elements before learning to speak.

5.3- We know that both geometry and arithmetics, as languages and sciences, developed after the integration between 1 and 2, originated at (4).

5.4- No doubts that expressions of period (1,2) are intellectual manifestations, their elements are thoughts (not behaviors, or instincts).

5.5- What I am introducing in the discussion is the old (somehow forgotten) argument by Socrates (exposed by Plato). He claimed that there should exist ideas (intellectual elements) which are more fundamental than others, at a level in which learning should not be needed. This is a line of thinking which considers the existence of `innate ideas'. In the modern times this line was defended by Leibniz, Kant, Jung and many others. Innate idea is a category a part, different from innate perception. To be innate in this sense does not imply that it has some genetic, or biological basis. Instead, it must be considered as reference a priori to express any biological, genetic or physical process.

5.6- So, considering this view, to establish the difference between algebra, topology, and other branches of mathematics, one should firstly establish a ground with objects which have sense by themselves. Then one can study the interrelations and external relations and define domains and etc. Relations do not necessarily cover all their intrinsic signification.

5.7- Of course there were and there are other views. During the lifetime of Leibniz polemics emerged on this matter.

5.8- Between Leibniz's death and the establishment of the universal algebra, there was a proliferation of concepts needing organization in fields. One of them was matrix.

5.9- Universal algebra emerged from a revision of logic, putting relations before `elements'. The name `algebra' was extended, this is OK, but does not help to answer the question of a possible conceptual hierarchy. In this universal view, meanings are given by relations, through relations, without proper ontology.

5.10- My point is that `universal algebra', `algebra of logic' and the revisions which were done, happened as reactions to the sudden resurgence of matrices in the modern mathematical arena.

5.11- Matrices had no origins of themselves, in spite of this, they were already showing themselves sticking out of the physical space, and were urging a physical position or status. Frequently, mathematicians were also physicists or were in close relation. Strange objects were shaking the physical grounds. Modern physics was approaching. And the threat of metaphysics was too big to be faced then.

5.12- The new logic was a salvation. Indirectly, the concept of vector was introduced as a very natural one, even interpreted as physically natural. But it did not exist before, appeared during this crises. They are not so `natural'. To exist, they always carry with them a shadow from a dual space! The current interpretation, which fits well in the philosophy of the operational physics, is that matrices are made of vectors.

5.13- At the turn of the 19th to the 20th century, the threat of metaphysics was openly discussed. Logical empiricism (mostly logicians and mathematicians) declared their position in a famous document, radically rejecting `innate ideas'. This trend was established and had a great influence on the scientific developments and methodology of the XX century. Einstein's relativity put down Euclidean geometry as innate and became a great support for this philosophy, or it was misinterpreted as such.

5.14- The rejection of a proper ontology for elements, making them pure relational, has an unifying operational appeal which is useful. But it has no compromise with physical demarcation, as it is necessary to keep track in the physical domain. It went out of control, leaving a huge price to be paid and the bill was given to physics alone.

5.15- Resulted from all that here we are, sharing our world with an apparently unrecognizable domain of `physical-matrices', as revealed by Heisenberg, Dirac, field theorists, gauge theorists, and so on.

5.16- But physicists do not accept neither recognize `reality' which is in their equations, unless the formulas arrive in 1 or 2 up there.

5.17- So, our present crises is even more serious, what `physical reality' means nobody can say anymore. Please have a look on this movie, which shows well the situation: All calculation and no play makes Vince a dull boy.

https://vimeo.com/159015802.

6 As previously commented, origins of concepts and knowledge can be approached from different views. Sometimes they are intermingled. And this is somehow due a freedom permitted by epistemology.

6.1- At a certain moment, I started to suspect that one of the problems (with us physicists) could be the rejection of `physical-matrices' in the `physical reality'.

6.2- To break matrices and push the pieces into where they do not actually belong, and then making them physical, could perhaps be our big mistake. They should be already intact in a basic level of observation.

6.3- To enter in such a discussion became a big complication too. The beyond-physical is unthinkable, or observable, in order to be discussed. Physical references are borders and there should be only a narrow region, between references and concepts, to localize the discussion. This is a question correlated to the mathematical one, because physical languages start at level (4).

6.4- So, I chose to dig in this pre-matrix basis, in the company of my student, who is a chess player, a person naturally familiarized with the intricacy of the matricial underground.

6.5- We were encouraged and inspired by Frege's work, in which he tells about the formation of numbers and arithmetics.

6.6- The idea was to find out a niche, which could have shaped the concept of matrix, corresponding to a pre-numerical matrix. This would correspond to missing physical reference, to be combined to others, forming observational languages.

6.7- We then have something to tell about a pre-matrix stage, assumed to be parallel to (1,2), the stage 3. This defines a period (1,2,3), in which there are elemental thoughts with signification. Our study was based on an experience which can be considered as a `regression beyond learning and language'.

7- Perhaps this exposition (too long, sorry) can answer the questions and explain the reason why I made a sharp difference between algebra and topology.

7.1- I consider that the name algebra applies mainly to a language which emerged during the period (4). As a mathematical concept, matrix was formed by interrelations at (4), when geometry, arithmetics and algebra had their beginnings.

7.2- After stage (4) many kinds of relation between 1,2 and 3, can develop. Hybrid elements were also be defined, to bridge the domains. Mathematics became a very complex science.

7.3- In my view, a hierarchical organization of concepts is necessary to understand physical observation and its evolution. The topic of observation in physics has been my first motivation.

Claudia's reply touches some fundamental questions. I'm not good at philosophy as my thinking is rather pragmatic. Here are a few of my simple-minded thoughts.

5.1- In fact, why did we connect shapes with numbers?

By "shapes" you probably mean length and surface. Expressing length (or distance) in terms of numbers comes naturally from a habit of comparing with something familiar, a "unit". We now call this measurement. It occurred in most early cultures. Measuring surfaces requires a more developed arithmetic (multiplication, or even approximation processes as used by Archimedes)  and it was probably motivated by the needs of agriculture or architecture in more advanced cultures.

5.11- Matrices had no origins of themselves, in spite of this, they were already showing themselves sticking out of the physical space, and were urging a physical position or status.

I'm afraid that the origins of matrices are quite banal. Centuries ago, there was no cheap writing paper but only expensive parchment. To solve several equations in several variables one used matrices as shorthand and performed operations directly on this scheme. The original economy of writing turned out to have mathematical advantages too. This is not exceptional: think of exponential notation (allowing broken or negative exponents) and its advantages through the rules of exponentiation. In my courses for computer scientists, I used to call this "the hardware of the pre-computer era".

6.7- We then have something to tell about a pre-matrix stage, assumed to be parallel to (1,2), the stage 3.

Continuing my story about matrices, one can see this notation as an extension of a much simpler one: tuples (pairs, triples,...). This corresponds to a fundamental habit of making priorities, using naming order, listing things. Matrices are a two-dimensional outgrowth of this: each equation has variables with a naming order x, y, z, ... and on top of this, equations in these variables are listed. It is structural notation that turned out to be mathematically useful and even physically meaningful.

Through @algebrafacts and the blog of David Richeson https://divisbyzero.com/2010/07/26/algebra-the-faustian-bargain/

I came across this great quote from Sir Micheal Atiyah. It is in Barry Mazur’s foreword to Tobias Dantzig’s book Number: the language of science (the 2005 reprinting of the 1930 classic).

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. —Sir Michael Atiyah, 2002

which drew the reaction from "Mugizi"

Reminds me of this famous quote attributed to Herman Weyl:

“In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.”

Interesting that many prominent mathematicians seemed to have a strong preference for geometry… (yeah, sample size of two …)