Why is a topology made up of 'open' sets? [closed]

Question

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 measurement, 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.

Answer 1

Disclaimer: There are many topologists here and they may not like the philosophical flavour of my answer :-)

I think it all starts with the end... actually with the notion of "end". Take an open interval (a,b). It is bounded, yet you cannot reach its ends! At first this may look weird, but then one realizes this weirdness is to be attributed to the mathematically exact observation of this object (mathematicians can distinguish between so many things like point, set of points, boundedness, boundary etc.). Encountering such "weirdness" only shows the strong need for an exact and abstract formulation of "having no end". The lack of ends we then call "openness". If we are looking for the best generalization of this concept, the first approach would be of course a set-theoretical one. And in fact, in the case of intervals it turns out that the property "having no ends" is inhereted into arbitrary unions and finite intersections. Any attemps to expand this lead either to contradictions to our basic example or to unjustified reduction in generality.

Answer 2

My answer will not be of philosophical nature, neither historical, but perhaps pedagogical.

I find Munkress' Topology a great book. Among other merits, because its introduction, which I summarize as follows:

1. You recall what a metric space is. Define open balls and subsequently, open sets. Prove that, in a metric space:

   1.1 The empty set and the total space are open sets.

   1.2 The union of an arbitrary number of open sets is an open set.

   1.3 The intersection of a finite number of open sets is an open set.

2. Recall what a continuous map between metric spaces is (the $\epsilon$-$\delta$ definition). Prove the theorem that says that a map between metric spaces $f: X \longrightarrow Y $ is continuous if and only if $f^{-1} (U) \subset X$ is an open set for every open set $U \subset Y$.

And you have a motivation for the definition of topological space and continuous map as well.

Of course this is not an historical explanation of how topological spaces arised, nor does it justify why you chose these properties of open sets in metric spaces and not others: "experience" has told us that these are the good ones. (For instance, if I'm not wrong, when Hausdorff first defined topological spaces included the property of being... Hausdorff among the axioms. "Experience" -and not an a priori argument- showed us that it could be interesting to work with non-Hausdorff topological spaces.)

Answer 3

Think of the half-open interval $(0,1]$ with the usual open sets (e.g. $(1-\varepsilon,1]$ is an open neighborhood of 1. Then modify the collection of sets considered "open" so that every open neighborhood of 1 contains some set of the form $(1-\varepsilon,1] \cup (0,\varepsilon)$. See if students understand that this modification in which sets are considered open also modifies the way in which the space is connected together.

Answer 4

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!