Why the axioms for a topological space are those axioms?

Solution 1:

The heart of it is that, while your intuition is that "continuity" is about distances - you first learn continuity in terms of $\epsilon-\delta$ proofs - it turns out that continuity is really only about open sets. Whether a map $f:X\to Y$ between metric spaces is continuous is entirely determined by the open sets of $X$ and $Y$. If you have two different metrics for $X$ which determine the same open sets, the set of continuous functions on $X$ are the same.

Since topology is the study of continuity, it makes sense to only care about open sets, then, not metrics.

The original definitions for "topology" had more rules about the open sets that made topologies "more like" metric spaces. But as mathematicians started playing with these things, they found that it made sense to ask questions about continuity in the cases where these rules were broken, too. So the definition was broadened to give the widest meaning.

A very basic example might be left-continuity. A function $f:\mathbb R\to Y$ with $Y$ a metric space is called left-continuous if $\lim_{x\to a-} f(x)=f(a)$ for all $a\in\mathbb R$.

It turns out, there is a topology on the real line, call it $\tau_{L}$, under which a function $f$ is left-continuous if and only if it is (topologically) continuous as a function from $(\mathbb R,\tau_L)\to Y$. (Note: $\tau_{L}$ has a basis the intervals of the form $(x,y]$.) Now, $\tau_{L}$ is a point-set topology, but it is not one that comes from a metric. So the notion of "left-continuity" is an example of an idea that fits the point-set definition of continuity, but does not fit the $\epsilon-\delta$ notion of continuity - you essentially need to redefine it.

The fact that left-continuity and right-continuity together is the same as "normal" continuity is a statement about the three different topologies. Somehow, the usual real line topology is a combination of these two other topologies.

It turns out there is something really deep going on here. Somehow, the open neighborhoods of a point in a topology contains a lot of information - what we call "local information" - about behavior of "nice" functions at that point.

Solution 2:

The notion of open set may best be comprised by "If $x$ is in the open set, then all points with $y\approx x$ should please also be in the set".

This makes $\emptyset$ and the space $X$ itself open automatically - either because there is no $x$ to check or no $y$ that could complain.

And still without specifying further what $\approx$ really means, it follows immediately that an arbitrary union of open sets is open again.

We could stop here, but these axioms alone don't make a good structure yet. Thus it is motivated to have a look at the "other" set operation, intersection. If we have two open sets (with possibly different interpretations of $\approx$, say $\approx_1$ and $\approx_2$) it seems to be a nice feature to assume that any two points that are witnessed in two ways as being not too different are not too different and vice versa, i.e. that the conjunction of $\approx_1$ and $\approx_2$ should make a valid $\approx$, so the intersection of two open sets should be considered open again. And there we are. One might consider some strengthening, such as arbitrary intersections, but as already the case of metric spaces shows, this will more often than not boil down to considering the power set of $X$ (or possibly with some points identified), so quite boring.

In hindsight, it is just the case that these axioms very nicely give us practically all the nice properties of metric spaces plus applicability to some interesting structures that do not allow a metric.

Solution 3:

I think the usual axioms for topology developed in several steps. First, people worked with Euclidean spaces and their subspaces, and the most important topological notions, like convergence and continuity, were developed in this context. When this is done, one can hardly help noticing that only the notion of distance, not the rest of Euclidean structure is needed. Furthermore, there are other sorts of "spaces" with reasonable notions of distance, leading, just as in Euclidean space, to reasonable and useful notions of convergence and continuity. In particular, one has the notions of uniform convergence of functions and (I believe somewhat later) $L_1$ and $L_2$ convergence. So people abstracted the relevant properties of distance and thus defined metric spaces. But there are some quite reasonable notions of convergence that don't fit into the context of metric spaces. Perhaps the most important one is pointwise convergence of functions (on uncountable sets like the reals). In other situations, it is possible to define a metric but the metric tends to look awkward compared with the notions of convergence and continuity that it supports. I'm thinking here of things like uniform convergence on compact sets (say for functions $\mathbb C\to\mathbb C$), or convergence of functions (on Euclidean space, say) together with their derivatives of all orders. Such situations lead naturally to the question whether one really needs a metric or whether convergence and continuity make sense in a broader (and perhaps simpler) context. The key observation here is that, when one defines convergence and continuity using a metric, the role of the metric is just to produce a notion of neighborhood. Once one knows what is meant by a neighborhood of a point, one can proceed without further reference to the metric. And neighborhoods in metric spaces have special properties that are never really needed for topological purposes. For example, each point has a nested sequence of neighborhoods, the balls of radius $1/n$ centered at the point, such that every neighborhood of the point includes one of these. It turns out that one doesn't really need either the nestedness or the countability of this sequence. So one can (and people like Hausdorff and Fréchet did) axiomatize what's really needed about neighborhoods. That essentially produced the modern notion of topological space (perhaps with a little additional information that one could regard as "really needed", like the Hausdorff separation axiom). The step from this "neighborhood" axiomatization to the currently standard "open set" axiomatization is, I believe, motivated only by a desire to make the axioms look as simple as possible. I would think of neighborhood systems as the "true" intuition underlying the notion of topological space, while the open sets provide a convenient way of describing and working with neighborhoods.