There are many application of topology

For example

The application of digital topology in image processing operations such as image thinning, border following, contour filling and object counting. And Photoshop programs

For example, trusses consisting of metal bars is sensitive of its structure in the sense of endurance and stability, and the structure may be optimized using toplogy theory to get more endurance and stability while keep less self-weight. This optimization is common in the design of mechanical components.

I suggest that you may have a look to introduction to topology pure and applied by Colin Adams and Robert Franzosa

I thank of your answers

Topology is used in Mathematical Modeling also.

@ Shanker, Is there reference?

Topology is a fundamental science in math, it has application from the most basic math disciplines such as set theory , functional analysis, nonlinear analaysys, algebra to very high end engineering such has control theory. what do you exactly look for from topology ?

There are a number of recent publications that provide applications of topology.

See, for example,

S.A. Naimpally and J.F. Peters, Topology with Applications. Topological Spaces via

Near and Far, World Scientific, Singapore, 2013, ISBN 978-981-4407-65-6,

with applications in cell biology, topology of digital images (from a proximity space

perspective), visual merchandising, microscopy, descriptive Hausdorff spaces,

camouflage filters, handwriting forgery detection, quotient topology in digital image analysis, grills in pattern recognition and fossil groupings, topological psychology, admissible covers in micropalaeontology, hit and miss topologies in population dynamics, local near sets in Hawking chronologies, and local patterns in weaving, local chronology star patterns and descriptive proximity patterns in tilings.

Another recent source of applications of topology can be found in

Jim Peters and Som Naimpally, Applications of near sets, Notices of the American

Mathematical Society 59 (4), 2012, 536-542.

More recently, very sophisticated parts of topology (characteristic classes) have found applications in condensed matter physics in the theory of the so called topological insulators.

I think this is a good question to familiar any one with applications of topology. Actually, I learned many things of your answers.

Please see the following post about topology and its applications.

It's going to be impossible to give you a full appreciation of what topology is, simply because it deals with mathematical ideas that are a lot more advanced than those you have seen so far.

but, here are some glimpses:

think of geometric shapes like a circle, a square, and an annulus ( the region between two concentric circles)

geometrically, they are very different. But there is something the circle and square have in common, a property not shared by the annulus. Start at a point inside the circle or square and draw any loop that remains in the circle or square and returns to the starting point. That loop will have the property that you can "shrink" it down to a point without any portion of the loop ever leaving the circle or square.

But in the annulus, it is possible to draw a loop that surrounds the "hole" formed by the inner circle. It will be impossible to shrink that loop to a point without dragging part of it into the hole, and thus leaving the annulus.

think about which geometric regions have this shrink-to-a-point property and which do not. You'll soon discover that the distinction between the two types of regions has nothing to do with geometric properties like angles, areas, distances, etc. We call this loop property a topological property.

So, loosely, the subject of topology studies a region by looking at the properites of that region that are independent of the geometric properties of the region.

but it gets more abstract than that. typically, the regions being studied aren't regions in the plane, except for textbook examples. They tend to be regions located in "topological spaces", which are very abstract, often multidimensional, even infinite dimensional!

As for applications, I think it is fair to say that most of the applications of topology are not directly to "real life." On the other hand, topology provides mathematical tools that are useful to applied mathematicians and to theoretical physicists when they do their work.

there is a theory among physicists that the universe is not just 3 dimensional, but that there are extra dimensions curled up in a complicated way on an infinitesimally tiny scale. Topology is used to describe this weird higher dimensional space we might be living in.

Topology is just a slick and very general way to define the notion of continuity and (slightly less general!) limits/convergence.

Continuity and limit processes are pervasive in mathematics and physics. If you study limits more closely you quickly come to the conclusion that what converges if you look at it from one point view need not converge from another point of view. The notion of topology allows you to express that. For example, you can approximate the diagonal of a unit square arbitrarily closely by zigzags parallel to the sides. However those zigzags always have length 2, so don't converge to the length of the diagonal sqrt(2). This "paradox" is quickly resolved if you use topology: the Zigzag's converge to the diagonal in the so called uniform ( = max distance) or C^0 topology. The length of curve involves derivatives however, so you need convergence in a different, stronger, topology having "smaller" open sets, for example the so called C^1 topology.

It is a highly non trivial fact that dimension is a topological property, i.e. can be defined using topological notions and is preserved under bicontinuous maps.

In abstract Mathematics it is useful to put topologies on things that look very discrete like the set of primes where a subset is open if it contains all but a finite number of primes. Surprisingly that really helps in using geometric intuition to reason about number theory.

In addition to continuity and limits/convergence, there are aspects of topology that lend themselves to applications, provided we agree on the notion of point. Recall that the initial concept of topology, for V.A. Efremovic, is the concept of point of adherence (isolated point) of an arbitrary set. Such points have location but no measurable content. For applications, we need non-abstract points (also called places by Martin Kovar) that have both location and measurable content. Assuming that we start by topologising a set of non-abstract points. To some extent, this is what E.H. Kronheimer proposed in his article published more than 20 years ago:

E.H. Kronheimer, The topology of digital images, Topology and its Applications 46, 1992, 279-303.

If we agree on topologising a set of non-abstract points, then we can start considering the features of non-abstract points and arrive at a digital topology with myriad applications. In addition to continuity, limit points and convergence, we can also start considering closure, dense subsets, compact sets, and so on in a particular application area.

Viswanath,

During the first part of the 1930s, Efremovic viewed topology as continuous geometry. The story about Efremovic's view of topology comes from Smirnov's 1952 paper on proximity spaces, translated into English by the Amer. Math. Soc. in 1964. This story about Efremovic's view of topology tends to reinforce what you have written.

Viswanath:

Here is the BibTeX for the 1952 paper on proximity spaces by Smirnov:

@article{

Smirnov1952,

author = {Yu. M. Smirnov},

Title = {On proximity spaces},

Journal = {Math. Sb. (N.S.)},

Volume = {31},

number = {73},

Pages = {543-574},

Year = {1952},

note = {{E}nglish translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 5-35}

}

I have attached a copy of this article to this message. Please let me know if see connections between your current research interests and the notion of proximity from Smirnov.

Viswanath,

Here is the paper by Smirnov.

Best,

Jim

Thank you much for this paper and your observations

I have always looked at point-set topology as a way of adding rigor to the idea of "nearness of points" . Of course the numerous off-shoots differential topology and algebraic topology serve other purposes. All together, from the point of view of mathematics, topology is more a construct needed to justify the use of various tools in higher mathematics, mostly geometry. So I have always thought of point set topology as something like the alphabets, used to frame the language of modern mathematics to materialise certain ideas.

I would also like to suggest the Topology book by Colin Adams and Robert Franzosa which discusses applications in a more literal sense or traditional sense, so to speak.

Vishesh,

Perhaps you will find my book Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces, Springer, 2014 interesting (see attached cover).

Another source of many applications of Topology is

S.A. Naimpally and J.F. Peters, Topology with Applications, World Scientific, 2013.

Dear Prof. Peters

Thank you very much.

it will be useful.

The best wish

@ Prof.Peters Thank you for that references and interesting book. Seemingly most modern day applications for topology involve discretization of some sort. Is that true??Or are there proper applications for the "continuous" ones??

Being a student of geometry, I see a lot of discretization going on in Geometry too

Please see the following

Qefsere,

Good post! Here is another source of applications of algebraic topology:

Applications of Algebraic Topology: Graphs and Networks. The Picard-Lefschetz Theory and Feynman Integrals (Applied Mathematical Sciences) [Paperback]

S. Lefschetz

See attached image.

I suppose you mean other than in mathematics, since there the applications are ubiquitous. In physics, there is the theorem: "the boundary of a boundary is zero" which according to Wheeler is to what the laws of physics ultimately reduce. (See _Gravitation_ with Misner and Thorn.) Examples are topological solitons, differential forms and de Rham cohomology, gauge theories and Aharonov-Bohm effect...

@ Vishesh Bhat: Seemingly most modern day applications for topology involve discretization of some sort. Is that true??Or are there proper applications for the "continuous" ones??

Yes, there are definitely applications for the continuous case. Prominent among these applications is the retract of a set X. Let Y be a subset of X.

Y is a retract of X, provided there is a continuous map r: X ---> Y with r(y) = y for all y in Y. The map r is a retraction. This is the basis for shape theory introduced by K. Borsuk, starting in the late 1950s and extending to 1980. For more about this, see pages 58-59 in the book by H. Edelsbrunner and J.L. Harer:

Book Computational Topology: An Introduction

See, also, A. Hatcher, Algebraic Topology, ch. 0, especially pages 1-3:

http://pi.math.cornell.edu/~hatcher/AT/AT.pdf

One of the most prominent applications of the continuous case is in shape theory. See Section 2, starting on page 482 in

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.468.2750&rep=rep1&type=pdf

and

https://www.elsevier.com/books/shape-theory/mardesic/978-0-444-86286-0

Also consider in Iran you have leading researchers in the mathematical chemistry field that provides many examples of applied topology.