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I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of many examples, but it's never been obvious to me how it came about, compared, for example, to the rather intuitive definition of a metric space. In some ways, the sparseness of the definition is startling as it tries to capture, apparently successfully, the barest notion of 'space' imaginable.

I can try to make this question more precise if necessary, but I'd prefer to leave it slightly vague, and hope that someone who has discussed this successfully in a first course, perhaps using a better understanding of history, might be able to help me out.

Added 24 March:

I'm grateful to everyone for their thoughtful answers so far. I'll have to think over them a bit before I can get a sense of the 'right' answer for myself. In the meanwhile, I thought I'd emphasize again the obvious fact that the standard concise definition has been tremendously successful. For example, when you classify two-manifolds with it, you get equivalence classes that agree exactly with intuition. Then in as divergent a direction as the study of equations over finite fields, there is the etale topology*, which explains very clearly surprising and intricate patterns in the behaviour of solution sets.

*If someone objects that the etale topology goes beyond the usual definition, I would argue that the logical essence is the same. It is notable that the standard definition admits such a generalization so naturally, whereas some of the others do not. (At least not in any obvious way.)

For those who haven't encountered one before, a Grothendieck topology just replaces subsets of a set $X$ by maps $$Y\rightarrow X.$$ The collection of maps that defines the topology on $X$ is required to satisfy some obvious axioms generalizing the usual ones.

Added 25 March:

I hope people aren't too annoyed if I admit I don't quite see a satisfactory answer yet. But thank you for all your efforts. Even though Sigfpe's answer is undoubtedly interesting, invoking the notion of measurment, even a fuzzy one, just doesn't seem to be the best approach. As Qiaochu has pointed out, a topological space is genuinely supposed to be more general than a metric space. If we leave aside the pedagogical issue for a moment and speak as working mathematicians, a general concept is most naturally justified in terms of its consequences. As pointed out earlier, topologies that have no trace of a metric interpretation have been consequential indeed.

When topologies were naturally generalized by Grothendieck, a good deal of emphasis was put on the notion of an open covering, and not just the open sets themselves. I wonder if this was true for Hausdorff as well. (Thanks for the historical information, Donu!) We can see the reason as we visualize a two-manifold. Any sufficiently fine open covering captures a combinatorial skeleton of the space by way of the intersections. Note that this is not true for a closed covering. In fact, I'm not sure what a sensible condition might be on a closed covering of a reasonable space that would allow us to compute homology with it. (Other than just saying they have to be the simplices of a triangulation. Which also reminds me to point out that homology can be computed for ordinary objects without any notion of topology.)

To summarize, a topology relates to analysis with its emphasis on functions and their continuity, and to metric geometry, with its measurements and distances. However, it also interpolates between these and something like combinatorial geometry, where continuous functions and measurements play very minor roles indeed.

For myself, I'm still confused.

Another afterthought: I see what I was trying to say above is that open sets in topology provide an abstract framework for describing local properties of functions. However, an open cover is also able to encode global properties of spaces. It seems the finite intersection property is important for this, but I'm not able to say for sure. And then, when I try to return to the pedagogical question with all this, I'm totally at a loss. There are very few basic concepts that trouble me as much in the classroom.

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closed as no longer relevant by Dan Petersen, Felipe Voloch, Mark Sapir, Bill Johnson, Andy Putman Jan 16 '12 at 5:00

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Do you find the Kuratowski closure axioms intuitive? If so, then the proof of equivalence between the Kuratowski closure axioms and the standard axioms is not hard. – Qiaochu Yuan Mar 23 '10 at 23:14
Maybe "environments" is what some of us call "neighborhoods" (and others of us call "neighbourhoods"). – Gerry Myerson Mar 23 '10 at 23:36
filters and nets are equally powerful logically, nets are useful because you can use your intuition of sequences in metric spaces (more or less), filters are useful because the statements about convergence become much shorter and prettier.. – jef Mar 24 '10 at 3:08
Regarding the effectiveness of the standard definition of "topology": I feel comfortable with its effectiveness in areas such as functional analysis and differential geometry. But I have never understood why the standard definitions of topology should be useful at all when working with finite fields and other discrete objects. Is there any way to motivate that for the non-expert? – Deane Yang Mar 24 '10 at 13:29
It's ironic that this question about open sets is now closed. – KConrad Jun 18 '14 at 21:38

35 Answers 35

I'm going to take a finite, "combinatorial geometry" approach, thinking about, for instance, convex polytopes.

Suppose we have a finite set of things. For instance, a tetrahedron is made up of 4 points, 6 lines, 4 faces, and one solid. We notice, first, that some of these things are adjacent to each other, and some things are not adjacent to each other. So we have a graph.

Then we further note that this graph can be directed. If two things are adjacent, one of them must be bigger. If they have the same dimension, we have forgotten about the boundary that, in fact, separates them. This is a consequence of including, in some sense, all the shapes you can, to maximally clarify the geometry of your space.

So we have a directed graph. It is easy to verify that this graph should satisfy the axioms of a partially ordered set. If $A \geq B$ ($B$ is an edge of $A$) and $B \geq C$ ($C$ is a vertex of $B$) then $A\geq C$ ($C$ is a vertex of $A$).

Now, finite partially-ordered sets are just finite T0 topological spaces. The open sets contain everything $\geq$ their elements, and the closed sets contain everything $\leq$ there elements. It is not obvious in this context why these are natural objects of study, although they are fairly easy to define, and thus must be useful for something.

To determine the difference between finite and infinite union, we must break out of the finite world. We're going to do that, however, only by breaking it up into smaller finite pieces. A face might turn into 4 faces, 4 edges, and 1 vertex, for instance. A set on the unbroken space becomes a set on the broken space. Its closure is preserved, while the smallest open set containing it gets smaller.

Thus, though in the finite case we can consider the smallest closed set containing something or the smallest open set containing it, only the first notion is preserved as we increase the number of objects in our space and decrease their size, bringing us closer to infinite, continuous mathematics. Thus, we abandon the notion of a smallest open set containing something, which means we must abandon infinite intersection of open spaces, and therefore, infinite union of closed spaces.

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I have long understood that the initial ideas of topology arose from the notion of "neighbourhood" and were then found to be equivalent to the definition in terms of open sets. One advantage of the neighbourhood concept was that the definition of continuity using that is nearer to the $\varepsilon$-$\delta$ definition used in analysis.

The neighbourhood definition is more easily motivated than that in terms of open sets, but one then shows the equivalence. However one finds difficulties with the neighbourhood definition in defining, say, identification spaces, and this illustrates nicely a feature of mathematics, that equivalent concepts may have their best uses in different areas. Horses for courses!

Einstein wrote in 1915:

"Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. They then become labelled as conceptual necessities, a priori situations, etc. The road of scientific progress is frequently blocked for long periods by such errors. It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little... "

Thus Grothendieck in his 1984 "Esquisse d'un programme" Section 5, argues that the notion of topological space is motivated from analysis rather than geometry, and the latter requires spaces with more structure, in particular what he calls stratified spaces. I have found filtered spaces important in basic homotopical algebraic topology.

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To me, like someone said before, it's because of the way you define continuity in a metric space... We've all heard (i think) that topology is a game "where bending and stretching is allowed but tearing is not", this is precisely what a continuous function does, so now, given a weird rubberish material (our topological space) how do we define continuity? Thinking back to metric spaces we find our $\epsilon$'s and $\delta$'s which are really just instructions on how to construct our space, they give us a definition of "nearness" (or "separation" some might say), so we construct our pieces of space with that definition in mind, we want a notion of when points are "near" (or "far") and we call those sets of relativily-near points "open"... To me it also helps to think of manifolds, like the usual example, the earth! it looks flat, but that's because we're looking at it locally, and what does "local" mean? an open set!

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I found that the comment box underneath andrews response wasnt large enough for what i had to say. I think that before i continue in my answer i should mention that i study homotopy theory, and maybe that is why i dont really care about motivating the "original" definition of a topological space. In homotopy theory, and perhaps any geometric flavor of topology, we work with things that have the homotopy type of a CW-complex, these may be much easier to motivate.

I think that the best way to motivate the definition in terms of open sets is historical (i think this is often the case, when you look at what people were thinking about or the problems they were trying to solve or overcome the definition might become clearer). When people started writing down the definition of what a topological space was there was a strong penchant for axioms and set theory. This is the flavor of the definition in terms of open sets. The definition that we have in terms of open sets was gotten after a bit of hard work with bad definitions. There was a lot of change in the culture of mathematics at the turn of the century and a lot of things had to be reworked and made rigorous. Perhaps i have the facts wrong, but it makes some sense this way even if i am mistaken.

One of my instructors frequently answers questions by saying things like we dont care about that or that is a bad question, which i feel is a legitimate response. The point is that there is a lot of mathematics to be done, a lot of really beautiful important mathematics. You can't really do all of it in a lifetime, so it is probably good to accept some simplifying assumptions like your ring is Noetherian or your space has the homotopy type of a CW-complex. The objects you are ignoring are not that natural to begin with and the things you are looking at are really much more important. In the end the questions we don't answer about the topologists sine curve won't really matter (...I think?) How could you hope to answer a question about some pathological special example with a tool that is meant to capture intuition?

since i dont know how to save this answer as a draft i will just have to settle for coming back to edit it later

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For me, "open" sets are called that because they are not closed. "Open" is the opposite of "closed". This is not a joke. The notion of "closed" comes from the idea that you can't escape the area following a sequence inside it. This captures what a closed set is.

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But of course, in mathematics, "not closed" is absolutely not equivalent to "open". That's the number one "rookie mistake" when these concepts are taught to undergraduates. – Spiro Karigiannis Jan 7 '12 at 12:24
Q: What is the difference between a set and a door? A: A door cannot be both open and closed at the same time. – Toby Bartels Jan 14 '12 at 7:01

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