There is a condition called 'local finiteness': suppose that every point has a neighborhood that intersects only finitely many of those closed sets. In that case, the union of the closed sets will be closed.

Hi Manuel,

I'm afraid my answer is not very interesting:

1. In a discrete space (a set X with the powerset P(X) as topology) this would be the case (not interesting). Spaces that satisfy the property that any union of closed sets is closed are Alexandrov spaces. There are different characterisations of such spaces.

2. If a T1-space contains limit points (which are necessarily \omega-accumulation points) one can always construct a countably infinite family of closed sets whose union is not closed. In a T1-space the property "that unions of closed sets are closed" is equivalent to the non-existence of limit points.

In a T1-space with limit points one might try to characterize the families of closed sets A_i whose union is closed: "For every limit point p there is an i with p\in A_i OR a neighbourhood U of p such that only finitely many A_i meet U."

[Remark, 25.4. : This property is (slightly) weaker than the "locally finite" mentioned below. Unlike local finiteness it covers e.g. the case of an exhaustion of a space by closed sets or the exhaustion of closed subset by closed sets.]

I'm not sure that there is an example where such a characterization is useful.

3. If you deal with countable unions of closed sets you might be content with knowing that the countable unions of closed sets are \( F_\sigma\) -sets and use the known properties of such sets. Or you might ask for conditions which guarantee that every F_\sigma-set is closed. This is precisely the definition of P-spaces, s. http://www.sciencedirect.com/science/article/pii/0016660X72900268

Yours, S.

There is a condition called 'local finiteness': suppose that every point has a neighborhood that intersects only finitely many of those closed sets. In that case, the union of the closed sets will be closed.

You are right - locally finite family of sets is a good answer - see Engelking, R. "General Topology", Helderman, Berlin, 1989. pp. 29-31

Hi everyone,

If the union of any family of closed sets is closed, at least whenever the space $X$ in consideration is $T_1$, then it is discret. For evey subset $Y$ of $X$ is the union of the closed sets $\{x\}$, where $x$ runs over $Y$. So there is no hope that such a condition be satisfied in interesting spaces.

Unfortunately the question not formulated clearly.

If the question is: "Are there topologies in which each unition of closed sets is closed" the answer is yes: all discrete topologies have this property.

(This is the interpretation which underlies Lahbib's respond)

If the question is: "In a given topology, are there systems of closed sets with closed union" the answer is yes: take any locally finite system of closed sets.

(This is the interpretation which underlies Daniel's and Mieczyslaw's respond)

Thanks to anyone of you for your answers.

Related to the answer of Dr. Mutze, which unifies the other answers while it is very precise as well , it seems that:

- A topological space X is discrete if and only if all all its subsets are clopen ( that is simultaneously open and closed )-

So, in this case, it is apparent that the countable union of closed -then also open- sets ( equivalently, the countable union of open sets) is open.

But the countable union of open sets is open in basic set theory , without the need of using complicated topological concepts/formulations.

Therefore, this is a very atypical case, although certainly it is a real one .

Manuele,

your statement shows that you did not fully understand what topology is about. 'Basic set theory' does not know the concept of 'open set'. Its the role of topology to introduce this notion.

When you explain to children of 12 years the concepts : open set, closed set, union of sets . Do yo explain these concepts with " topology"?

Manuel,

of course, union of sets is for 12 years old children. But what do yo tell them about open and closed sets? Saying that you call a set open if with any point there belongs also some small interval (or, disk, or shere) to it is just explaining topology in a simple special case. (I share your uneasiness about considering savant sounding word such as 'topology' an essential part of such an explanation). A 12 years old child will probably be able to understand and memorize such an explanation. It very probably will not understand the reasons that made adults consider such stuff.

Dear Ulrich:

I agree with your points . However, what I wanted to express is that the distinctions between kinds of topologies , metrics etc. are introduced in Advanced University Courses, mainly in careers of Mathematics and Theoretical Physics. The concept of ball , its radius etc. is , of course, essentially topological and metric but it is also intuitive for non-specialists and it is what is addressed in most of courses even at University . So the " euclidean" is intuitive but most of "the others "are not.

The concepts of discrete topology, discrete metrics, weak, strong , weaker than , stronger than etc. topology are teached in advances courses of mathematics ( for instance , using books like the well- known Rudin book) , not in the basic courses, except may be for some small introduction.

So , the basic general principle without entering in sophisticated explanations ( if you are responding me later on that what follows also requires " topology", I agree since now) , is that the infinite union of open sets is open, the infinite union of closed sets can or can´t be closed, the infinite intersection of closed sets is closed and the infinite intersection of open sets can or can´t be open. Those principles appear in basic books of mathematics given in the first course of University to all sciences careers.

In the discrete topology, the sets are simultaneously open and closed ( I think) so that the infinite union of closed sets is equal to the infinite union of open sets, so open, but also closed, since we are with the discrete topology. But , I think we have to consider this as an exception to the general rule for infinite union of closed sets( or , dually, for infinite intersection of open sets) , the use of the discrete topology is not usual , to my knowledge, in application-related problems, may be it can be used in coding theory or related fields.

But , anyway, I fully agree with your first precise answers to your reformulated ( and more precise ) questions to my previous one in terms of " are there topologies...?" and " in a given topology..."?.).

A necessary and sufficient condition is certainly that the topology

can be derived from a single relation. (See perhaps a paper of G. Birkhoff

or our papers relators.)

@Arpad

What is ? Please explain >. It is not difficult to characterize any topology by a single relation: Consider a set X and a subset T of the powerset P(X) which is a topology. Then define a relation R on P(X) by aRb := a=b and a element of T. The a belongs to T (is open in topology T) iff aRa. Thus the topology T is derived from relation R. Certainly G. Birkhoff considers something much more interesting. It would be great if you could tell us.

Dear Manuel,

in my mathematical education the discrete topology (i.e. T = P(X) in the notation of my previous contribution) played some role as a handy device for testing conjectures. It also made me wonder that an axiom system that allows such a simple model (it has an even more simple model, the trivial topology) can bring about such nice and sophisticated structures.

Dear Ulrich, your example in the response to Arpad, is very interesting.

Concerning an example of the power set P(X), where X ={a, b, c ,..} ( a finite number of elements) , I am wondering it if can be descibed in the following dual way:

1- Characterize the power set P(x) of X as a set of closed sets with empty interior so that all elements belong to the boundaries of some of the sets of P(X) . Those sets (with empty interior ) are closed since the boundaries of sets are closed sets.

2- The empty set and P(x) are closed. Here the empty set adopts the role " closed set" as being the intersection of any set of disjoint closed sets.

3- Built a topology of closed sets . Each member of the topology used to characterize " closed sets"( which are (closed boundaries of sets since boundaries are always closed ) with empty ( closed) interiors) has to contain at least one element of the interior ( this is easy since the interior is empty, which is a subset of any set, + at least one element of the complementary subset in P(x)) . This seems to be easy since the elements of the power set contain common elements between the various subsets.

4- The binary relation can be the same as that of your example of the discrete topology (i. e. any two elements are related by the relation R if they are identical) .

5- So, we have a dual example based on a topology of closed sets which have nonempty ( closed ) boundaries and (closed ) empty interiors. The topology consists of "closed" sets based in the same relation as that of the example.

1-By the way, are the open sets ( closed here if I am right ) of the topology in your example are just the elements of P(X ) ?.

2-So, in this kind of topologies, is it the same to speak about open or closed sets just depending only on the way of defining the " structures" and " natures" ( open versus closed) of the all sets, including the empty set?.

[I am supporting this statement based on the fact that the literature which considers the dual way of defining topologies with closed sets is ,not very common, but established].

Dear Ulrich:

I do not like very much very weak topologies as those we are using in these comments.

Assume that you are just at the top-point of the Everest , you have to fix the next point of your walk while you are characterizing your motion trajectory via a so weaker topology that your trajectory is a continuous map at the top- point......

Dear Manuel: our mathematical educations seem to have happened on different planets.

For me the discrete topology on a set X is the strongest among all topologies on X and not a 'very weak one'.

I can't agree with you that my example @Arpad is interesting. It only shows that what naturally is a predicate can be made a binary relation in a trivial (opposite to interesting) way. How can you characterize the power set of a finite set in a way that notions of 'boundary' and 'interior' are employed?

Your Everest example does not make sense to me in any of the few interpretations that I tried. I guess I would need the influence of low oxygen pressure to understand it.

Dear Ulrich ,

According to your advices , I would take an extra oxygen tank in my next trip to Everest just in case I have some problems at the top with function continuity from a sufficiently weak topology.

Related to the example of the discrete topology on a set X, I still think it is interesting to describe the flexibility of defining topologies.

I agree that the proposed topology is the stongest one for that problem . How to define a weaker one and the weakest one?.

The concetps of boundary and interior , i referred to the boundaries and interiors of the sets taking part of the power set of the initial set. For intance, typically and for sets of just isolated points ( may be I was not very clear in this sub-example of yours, I omitted that X was a set of islotated points) , it is said that the sets are open so the points defining the sets are in the interior set. But , is it possible to say that the points are in the boundary and the interior is empty. For instance {x} (x ,a point ) is an open set and it is equal to its interior. Could this be reformulated in terms that that the set is closed , equalizes its boundary, which is of course, closed as well and the interior is empty?.

Dear Manuel,

unfortunately none of the sentences in your last contribution makes sense to me. So, I see no sense in continuing this discussion. Our mathematical worlds are separated by too large a distance.

Dear Ulrich,

Thank you very much for your remarks.

You are right. There are several more or less

natural possibility of deriving open sets from relations

or relators (families of relations).

I had in mind the most natural one. If R is a relation on X, then

the singleton { R(x) } is a neighbourhood system for every x in X.

Equivalently, cl(A)=R^{-1} [A] is the closure of a subset A of X.

Thus, the set A is closed if and only if cl(A) is a subset of A.

In my remark, I wanted to write that "see our papers on relators",

or rather that on simple relators, which include the weell-chained, and

the connected ones.

Perhaps, the following papers clarify the developments.

1. G. Birkhoff, Rings of sets, Duke. Math. J. 3 (1937), 443--454.

2. G. Pataki, Supplementary notes to the theory of simple relators,

Radovi Mat. 9 (1999), 101--118.

3. G. Pataki and A Szaz, A unified treatment of ....,

Acta Math. Acad. Paedagog. Nyhazi (electronic)

19 (2003), 101--165.

4. A. Szaz, Upper and lower bound in relator spaces, Serdica 29

(2003), 239--270.

5. A Szaz, Minimal structures, .... , Filomat 21 (2007), 87--97.

6. A Szaz, Lower semicontinuity properties of relations in relator

spaces, Adv, Stud. Contemp. Math. 23 (2013), 107--158.

With best regards,

Arpad

Dear Arpad,

you place me on the school desk.

>.

Let me, for absolute clarity, use an unabridge notation: (x,y) \in R for xRy.

Am I right in interpreting your R(x) as {y \in X | (x,y) \in R } ? (BTW, I would write this as

R[x]). R(x) would then be a subset of X. How than can the single elemented object

{R(x)} be a > ? Am I blind or did you mix something up ?

Best regards, Ulrich

Dear Ulrich,

Yuor understanding of R(x) for a point x of X of is correct.

To understand this notation, note that here R can, in particular,

be a function. Moreover, RA] has to be reserved for the image ,

under R, of a subset A of X.

To understand the further criticized part of my remark,

you may equivalently consider the filter on X generated

by { R(x)}.

Otherwise, the proper definitions of most topological and uniform

notions (such as continuity, well-chainednes, connectedmess,

convergence, Cauchy property, compactness, and

Baire property, for instance) can only be given in relator spaces.

For instance, a net (x_{\alpha}) in a relator space X(\Cal R) has

to be called Cauchy it converges in each of the properly simple

relator spaces X(R) where R runs in \Cal R.

To realize the appropriateness of this definition, whose origin

goes back to Cohen (1937), Konishi (1952) and

Siber--Pervin (1965), take a metric d on X, and consider the

relator R_{d} on X consisting of all r-sized surroundings

{ (x, y) : d(x, y)

Dear Ulrich:

I think we are missing to give a relevance to the empty set in the discussed topologies. Therefore , I think that we have to include it in the topology definions ( as an open set if the topology is defined with open sets or considering it as a closed one if the topology is defined with closed sets). Instead, may be, we could omit it in an explicit definition of the top. while , instead, to give some axiom related to it in the topology definition.

Dear Arpad,

Aha! You consider the topology generated by the system {R(x) | x \in X} of sets. Your words suggested something different to me. This construction reminds me at the initial and final topologies generated by a family of functions. These are of great organizing value in functional analysis.

Thanks for clarification.

Dear Manuel,

where did we mistreat the void set?

Dear Mutze,

Tis is also a natural topology derived from the relation R.

However, it greatly differs from that I was writing about.

Sincerel yours,

Arpad

Dear Arpad,

since all my attempts to understand

failed, I ask you for help, preferably in the form of a fully explicit unambiguous definition.

Thank you,

Ulrich

Dear Arpad and Mutze,

The problem with this topology generated by the equivalence classes with respect to an equivalence relation $R$ is that it is never $T_0$ except in case $R$ is antisymetric and than equality. But in this it is trivial (i.e. disrete). Two equivalence classes are disjoint or equal.

Dear Ulrich,

In other formulation, if R is an arbitrary relation on X

then for any point x in X and subset A of X we write

x \in int_{R}(A) if R(x) is a subset of A.

Thus, the set is called R-open if A is a subset of int_{R}(A) .

That is, R[A] is a subset of A.

A similar definition can be applied when you have a family of

relations on X, or more generally on X to Y. However, in that

gereral case the closure of a set can be expressed more

explicitly than its interior.

Otherwise, you shuold see the definitions of one of our papers listed

earlier. For me, it is a crazy thing to base on open sets and to

study generalized topologies and minimal structures.

Namely, such things can always be derived from relators

(families of relations).

Sincerely yours,

Arpad

Dear Lahbib,

In connection with your remark, I can only note the followings:

1. Every topology on a set X can be derived from a family of

preorders on X according to W.J. Pervin's observation.

2. If a remember well, topologies derived from an arbitrary

family of equivalence relations has been first studied by

N. Levine.

Sincerely yours,

Arpad

I am not so much answering the question which was originally asked, but instead providing an avenue by which you may obtain some additional information:

You are asking for conditions which imply that an infinite union of closed sets is closed. A space for which infinite unions of closed sets is closed is equivalent to a space for which arbitrary intersections of open sets is open. Such a space has a name: an Alexandrov space. You may want to check the literature (even wikipedia has a nice entry on such spaces).

Dear Greg Oman:

Thanks for this useful information concerning the initial question.

A P-space is one in which countable unions of closed sets are closed.  By modification of the definition, you can specify when unions of at most k closed sets are closed.  In my 1974 thesis I noticed that topological ultraproducts , via  countably incomplete ultrafilters, are P-spaces; if the ultrafilters are k-regular, unions  of k closed sets are closed.   A P-point in a space is a point x such that each  intersection of countably many neighborhoods of x is also a neighborhood of x.  A space is a P-space iff each of its points is a P-point; the existence of P-points in the Stone-Cech remainder of the integers is both consistent with and independent of ZFC. There's a huge literature along these lines.

Dear  Paul:

Many thanks for your answer. After a quick checking , it seems that the P-spaces you mention include Alexandrov spaces and also locally finite spaces which have also the property that  the infinite union of closed sets is closed. So , it seems that  this framework ( P-spaces)  is the most general related to the property.