Why the axioms for a topological space are those axioms?

Question

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard to motivate. Well, the ideia of a metric space is much easier to motivate: "the concept of distance is context dependent, so we want a general idea of what distance is and a general idea of a set on which we can measure distances". The axioms then for a metric are very intuitive, easy to motivate and everything else.

Then we start studying properties of subsets of metric spaces. We define open balls as a way to make precise the notion of "the set of all points that are sufficiently close to a central point" and we define open sets to make precise "sets such that for every point, other points sufficiently close are also in the set", which can also be thought as sets such that each point can be oscilated a little bit from it's position and the point will stay on the set.

After that we can define lots of things: limit points, closed sets, dense sets, perfect sets, compactness and so on. We also see that all of those notions can be made precise mentioning the open sets alone: the metric is not really necessary to talk about those things, as soon as we can define what open sets are. So this is enough motivation to define a structure on which we have open sets.

The answer to this problem is to define a topology on a set $X$ as a set $\mathcal{T} \subset \mathcal{P}(X)$ such that $\mathcal{T}$ is closed under arbitrary unions, finite intersections and such that $X,\emptyset \in \mathcal{T}$. If, $X$ is a metric space, and we let $\mathcal{T}$ be the set of open sets as they are defined using balls, then the three properties are satisfied.

My only question is: why those properties capture completely the idea of an open set? I mean, amongst all properties of open sets, why do we select those three? I've always heard that topology is meant to study qualitatively global properties of forms, that's the way we start in $\mathbb{R}^n$ and the way we generalize to metric spaces: we introduce tools that allows us to define carefully some of these properties and we work out definitions. It doesn't seem clear the connection of this motivation for topology and the actual definition.

I've seem a similar question on MathOverflow, and there was one answer trying to motivate this in terms of rulers, but I really didn't get the idea. Can someone give a little help with this?

Thanks very much in advance!

Answer 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.

Answer 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.

Answer 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.

Answer 4

I agree with Andreas.

The open set axioms for a topology are usually presented as the axioms for a topology, but in fact there are as number of equivalent ways of defining a topology, via neighbourhoods, open sets, closed sets, closure, and even interior or boundary. The neighbourhood axioms are surely the most intuitive to a beginner, and most clearly related to the usual notion of continuity in analysis. Indeed the definition: "$f$ is continuous at a point $x$ of its domain if for all neighbourhoods $N$ of $f(x)$ there is a neighbourhood $M$ of $x$ such that $f(M) \subseteq N$" is more direct and geometric than the $\epsilon$-$\delta$ definition, since the latter are just measures of the sizes of neighbourhoods, and so a step away from the geometric picture of a neighbourhood. Those numbers $\epsilon, \delta$ are of course a useful computational representation, as numbers are.

Of course the open set axioms are very elegant and logically simpler than the neighbourhood axioms, and are in some case the best way of defining a topology, for example for identification spaces. But beginners should be encouraged to make their own judgement on the advantages and disadvantages of each definition of a topology.

Part of the need for such abstraction from the notion of metric space was the development of ideas of Riemann surfaces, and then the notion of manifold. The work of Hausdorff was a key in this development.

January, 2017 I recommend here the Introduction to Peter Freyd's Abelian Categories part of which states: "If topology were defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology......A better (albeit not perfect) description of topology is that it is the study of continuous maps;..."

This reminds me of a quotation of Einstein: "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. $\ldots$ 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 $\ldots$" Difficulties in algebraic topology with the notion of "bare" topological space, i.e. with no extra structure, are discussed in this paper.