As an undergraduate I learned point-set topology from Munkres's book, as did many others.
One topic that gets a lot of attention is the separation axioms. For example, a space $X$ is normal if any two closed, disjoint subsets of $X$ can be separated by open neighborhoods.
Some of the axioms (e.g. Hausdorff) turn up a lot, but I feel like I virtually never hear topological spaces described as regular or normal (and there are a host of other properties too!) However, there are some fields I don't interact with too much.
In what contexts outside of general topology and set theory do these axioms, and the theorems you prove from them (Tietze extension, Urysohn's lemma, etc.), play an important role?
Thank you! -Frank