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](http://en.wikipedia.org/wiki/Separation_axioms) 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



Where else do the (topology) separation axioms turn up?
These separation axioms play an important role in determing duals of spaces of continuous functions (i.e. in functional analysis). Examples are: 


* If $X$ is normal, then the dual of the space of bounded real-valued continuous functions $C\_b(X)$ is the space of regular bounded finitely additive measures on $X$.
* If $X$ is completely regular Hausdorff, then the dual of $C(X)$ (continous real-valued functions on $X$) in the compact-open topology is the space of all Borel measures on $X$ with compact support.
* If $X$ is normal, then the dual of $C(X)$ in the pseudocompact-open topology is the space of all regular Borel measures on $X$ with pseudocompact support.


In the normal case above, the Tietze extension theorem is used to extend continuous maps from closed subsets to $X$. Similarily, in the completely regular Hausdorff case, a corresponding theorem is used to extend continous maps from compact subsets to $X$. 


