TITLE: Motivation for separation axioms
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: 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?"