Motivation for separation axioms

Question

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am namely talking about the $T_{k}$ and $N_{k}$ hierarchy.

I understand the Hausdorff property, i.e. $T_{2}$-property, is important in analysis and limits; and second countability, i.e. $N_{2}$-property, is important for example in integration on smooth manifolds when one constructs partitions of unity. Moreover, a metric space that is $N_{2}$ satisfies automatically most of the separation and all of the countability axioms. So what is the motivation of such a careful examination of all the different levels of separation and countability in general topology? Is it just for the sake of its own interest, or are there other good examples when one really needs to use different levels of this hierarchy? Especially the $T_{k}$ hierarchy.

Thanks a lot in advance.

Answer 1

I don't have a great deal of familiarity with the separation axioms, but I thought you might find it reassuring to learn that there are important examples of topological spaces that do not lie at the nice end of the $T_k$ scale.

1. The Zariski topology. Modern algebraic geometers study spaces of spectra of rings, whose points are prime ideals under the Zariski topology. Spectra are always $T_0$, but the interesting examples are usually not $T_1$. If you know some algebra and are interested, I can elaborate.

2. Finite topological spaces. As soon as a finite topological space is $T_1$, it is discrete. The finite topological spaces that are $T_0$ are in bijection with finite partially ordered sets. Specifically, every poset can be given a topology arising from its ordering, and the topologies on finite $T_0$ spaces always arise in this way.

With that said, mathematicians tend to prefer spaces on the high end of the $T_k$ scale. It is often the case in practice that one would like to understand a certain object using topology, and one has a choice of various natural topologies. The golden dream is to end up with some nice space, like a compact $T_2$ space; this is what happens with the Krull topology on a Galois group, for instance.

Answer 2

They are useful in different situations. For example, $T_{4}$ is of course stronger and "better" than $T_{3}$, except for the fact that $T_{3}$ is hereditary and preserved under product. So you may take advantage of working in a $T_{4}$ space through knowledge that the space is also $T_{3}$ and so every subspace of it is $T_{3}$ too, if we didn't have separate concepts for $T_{3}$ and $T_{4}$ we would be missing out on this information we have about subspaces of $T_{4}$ spaces.

P.S. A metric space, second countable or not, satisfies all the separation axioms. So effectively the separation axioms are only interesting when working on non-metrizable spaces. Maybe you'd like to change your question to "Why study non-metrizable spaces?"