Where else do the (topology) separation axioms turn up?

Question

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

Answer 1

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 $X$.

Answer 2

The property of normality turns up in the ideal theory of C*-algebras and related topics. If $A$ is a separable C*-algebra, or more generally if $A$ is $\sigma$-unital, then the complete regularization of the primitive ideal space of $A$ is $\sigma$-compact and hence normal. This fact has been exploited by several authors recently, and in at least two cases properties have been shown to hold when $A$ is $\sigma$-unital which do not hold for general $A$. For example, Aldo Lazar has shown that two natural topologies coincide on this complete regularization space if $A$ is $\sigma$-unital but not in general; see 'Quotient spaces determined by algebras of continuous functions', Israel J. Math., 179 (2010) 145-155.

For me, the most useful property of normal (Hausdorff) spaces is that disjoint sets have disjoint closures in the Stone-Cech compactification.

Answer 3

I highly recommend Chapter 16 of Eric Schechter's Handbook of Analysis and its Foundations (AP, 1997), which deals with separation axioms and their individual importance in quite a bit of detail. One reason that separation axioms are implicit (rather than explicit) in many mathematical discussions outside general topology is that basic topological spaces (such are $\mathbb{R}^n$) that we use to construct other spaces already satisfy all most of the separation axioms, and these properties survive these constructions so do not need to be checked separately (key words here are hereditary, productive, initial).

If you squint at some of the separation axioms, you'll notice they mostly have to do with pairs of points and pairs of open/closed sets containing them. Thus, separation properties of a topological space $X$ can be naturally expressed as topological properties of the Cartesian product space $X\times X$ and its diagonal subset, $X\to X\times X$. Recall that relations on $X$ are subsets of $X\times X$, so one might expect some non-trivial interplay between relations on topological spaces (like orders or equivalence relations) and separation axioms. For instance, some separation properties fail to survive the quotient of a topological space by an equivalence relation. Then they have to be discussed explicitly. For example, famously the space of leaves of a smooth foliation of a manifold may fail to be Hausdorff.