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By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms of open sets.) But I think in the practical cases, we just considered Euclidean space most and the traditional form open set in Euclidean space works pretty well. And these new form open sets seem to have no common use. So why we need to expand the conception of the open sets? What's the motivation behind?Thanks.

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I think your idea that topology is just to define open sets misses the point. I interpret the point of topology as attempting to generalize the concept of a continuous function primarily, and open sets are often the most direct pedagogical route there.

Further, one often does want to leave euclidean space. For example, we often want to consider spaces of continuous functions where the topology (say, pointwise convergence) may not even be metrizable! Topological spaces that are not first countable, and thus not metrizable, also arise quite naturally in functional analysis.

Even at times when one can fundamentally imbedd the space being worked on into some euclidean space, this is often a cumbersome process adding on unnecessary baggage.

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  • $\begingroup$ Thanks,but why do you think "attempting to generalize the concept of a continuous function primarily".In my mind, the definition of topology just tells you what are open sets and their characteristics. $\endgroup$ – 6666 Aug 31 '14 at 2:46
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    $\begingroup$ There's a few reasons. One, generalization and abstraction allow one to focus on the core of what is going on and remove unnecessary baggage. More importantly though, spaces show up that are not even metrizable in practical situations. For example, pointwise convergence of functions in function spaces. It would be nice to have a way to discuss this convergence but we can not use our metric space notions as is. Open sets on their own are really not what topology is about, they are important because they let one talk about continuity and limits. $\endgroup$ – GWilliams Aug 31 '14 at 2:50
  • $\begingroup$ Thank you very much, it looks reasonable, but a little abstract, could you offer me a concrete example? $\endgroup$ – 6666 Aug 31 '14 at 3:04
  • $\begingroup$ Here, this thread has a number of examples! mathoverflow.net/questions/52032/… In each instance the space is not metrizable, but we still want to discuss convergence and limits. $\endgroup$ – GWilliams Aug 31 '14 at 3:22
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For starters, we have things defined abstractly (at least in principle) such as manifolds. These are defined through charts, which endow your manifold with a topology.

The important tool of partitions of unity (which reduces Stokes's theorem and others to a toy case) works due to the 2nd Countability Axiom (see e.g. Warner, Foundations of Differential Manifolds and Lie groups).

There are important results in Commutative Algebra such as finiteness of irreducible components of the spectrum of a noetherian commutative ring: topology furnishes a proof of an algebraic result (see Atiyah & Macdonald, Introduction to Commutative Algebra).

Also, the topology is kind of qualitative, i.e. you are not considering a particular metric only, but only those equivalent to it (i.e. those which yield the same topology).

There are many more things one could say, but we'll leave it at that until further comments appear.


Where you can see a nice interplay between topology and analysis is in the Gelfand-Naimark theorem: the maximal rings of the ring of (real or complex) continuous functions over a compact topological space $K$ is homeomorphic to $K$ itself.

Again, a good reference is Atiyah-Macdonald (it's in the Exercises section, if I remember rightly).

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  • $\begingroup$ I have just added a comment on a well-known result. I think the Gelfand-Naimark theorem is a generalisation. $\endgroup$ – Theon Alexander Aug 31 '14 at 2:47
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We need topology so we can work limits. And limits are important.

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  • $\begingroup$ What kind of limits we can't work without topology? $\endgroup$ – 6666 Aug 31 '14 at 1:58
  • $\begingroup$ Excuse me? What does this question even means? The definition of limit is based on topology, the definition of converging sequence may be used to define a topology, and in general we can even define a notion of convergence to be equivalent to the definition of topological space. Topology is the idea behind closeness, and practically everything in analysis is about approximation. $\endgroup$ – Lolman Aug 31 '14 at 2:07
  • $\begingroup$ Thanks, you said we need topology to work limits, but topology defines different forms of opens, do we need these different form open sets to work on limits? $\endgroup$ – 6666 Aug 31 '14 at 2:12
  • $\begingroup$ It depends on the problem at hand, but yes. If you want to be able to speak of limits you need first a topology. (Or if you want a topology you may define what converges where.) $\endgroup$ – Lolman Aug 31 '14 at 2:18
  • $\begingroup$ Here there are issues: one may have two limits to a sequence, if we work with a non-Hausdorff topological space. $\endgroup$ – Theon Alexander Aug 31 '14 at 2:48

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