Importance of Locally Compact Hausdorff Spaces

Question

I mostly deal with measure and probability theory and quite often, whenever I look up something on wikipedia, I see the mathematical objects defined on a locally compact Hausdorff space.

I have very little background in topology and while I do understand the definition, why do I see this space so often is something that you can't simply see from the definition itself.

My guess is that it is in some sense a generalisation of the spaces we deal with (say $\mathbb R^n$), which is general enough to include a variety of spaces, but restricted enough to keep the nice properties we want. Similar to, say, formulating results in analysis in a metric space (even if we're mostly interested in $\mathbb R^n$ or even $\mathbb R$), or probability results formulated in $\sigma$-finite spaces (even though we really have a finite space).

Therefore: is the guess above correct? If so, what are some of the nice properties? Is there a particular connection to probability theory?

I would consider answering the first question sufficient, but would very much welcome a context along the lines of the second and third question.

Thank you.

Answer 1

I know little topology, but since people aren't answering, I thought I would share at least some thoughts.

I think your guess is correct (though with how it's written, I'm not sure whether a guess like that could be wrong).

Paraphrasing Topology - James R. Munkres, two very well behaved spaces that we can hope to work with would be metrizable spaces, and compact Hausdorff spaces. If a space is not one of those, then what we could hope for, is that it is a subspace of one of those. Since a subspace of a metrizable space is still metrizable, that's not really something we have to consider, but a subspace of a compact Hausdorff space does not have to be a compact Hausdorff space.

It turns out, that a locally compact Hausdorff space which is not itself compact, is a subspace of a compact Hausdorff space that only has one additional point in it, and it is also dense in this space - this compact Hausdorff space is called one point compactification, and it is guaranteed to exist for any locally compact Hausdorff space (and to be exact, existence of one point compactification for some space $X$ is equivalent to the space $X$ being a locally compact Hausdorff space).

There's also the fact, that for any closed or open set $A$ in a locally compact Hausdorff space, $A$ itself is locally compact Hausdorff subspace.

A bit more information about the one point compactification:

https://en.wikipedia.org/wiki/Alexandroff_extension

Answer 2

I think it is probably more fruitful to ask what is special about the category of compact Hausdorff spaces. This is not much of a restriction, for the following reasons:

• Every locally compact Hausdorff space is an open subspace of some compact Hausdorff space, e.g. the one-point compactification. Note that the one-point compactification of a Hausdorff space need not even be Hausdorff without local compactness.
• Every open subspace of a compact Hausdorff space is a locally compact Hausdorff space. Note that this requires the Hausdorff assumption, since open subspaces of compact spaces are not locally compact in general.

There are a number of nice things about the category of compact Hausdorff spaces. For example, there is the Banach-Stone Theorem, which says that a compact Hausdorff space can be reconstructed from its topological algebra of continuous functions. In fact, there is a very long list of nice properties of continuous functions on compact Hausdorff spaces.

More concretely, compact Hausdorff ($T_2$) spaces are also normal ($T_4$), which means that we have various nice tools like the Tietze extension theorem that allow us to connect our space to the real numbers in concrete ways.

Then there is the Stone-Čech compactification, which gives a well-behaved, universal (i.e. left adjoint to the forgetful functor) way to transform any topological space into a compact Hausdorff space. In fact, buried in this construction is the idea that any compact Hausdorff space can be obtained by weakening the standard topology on $[0,1]^S$ for some set $S$.

We might sometimes want to work in a smaller category, for example the category of compact metrizable spaces. But this turns out to be exactly the same category with the additional axiom of second-countability. And if we don't need second-countability for our theorems to hold, then we might as well do without it. The same goes for many such attempts to restrict study to categories in between Euclidean spaces and compact Hausdorff spaces.

I'm certainly leaving out a number of things, but perhaps this gives some idea of why this particular category of spaces tends to be a highly privileged one.