# Are there useful categorical characterisations of the topological separation axioms?

Tietze's extension theorem states: If $X$ is a normal space, and $A$ a closed subspace. Then any continuous function to the reals $f:A\rightarrow R$ has an extension to $f':X\rightarrow R$ that is $f=f'i$ where $i$ is the inclusion of $A$ in $X$.

This can instead be used to characterize normal spaces. With this definition there is an easy proof of Urysohn's Lemma.

Essentially this is a categorical characterization of Normality. Are there similarly nice & useful categorical characterizations of the other separation axioms?

There is also one for Tichonov spaces - these are completely regular $T_1$ spaces (in fact it always turns out that they are $T_2$). They always have Hausdorff compactifications (a Hausdorff compact space within which they embed densely) and this characterises them. Since there is always a maximal compactification - the Stone-Čech compactification. This gives a categorical charactisation of Tichonov spaces.

• You really ought to specify: Tikhonov spaces always have Hausdorff compactifications. (For some of us a compactification is simply an embedding as a dense subset of a compact space, and compact spaces are not necessarily Hausdorff.) Jul 2, 2013 at 8:32
• @Scott: ok, done. It ought to have been Tikhonov spaces rather than completely regular spaces. Jul 2, 2013 at 11:24

What a funny coincidence, I've just attended a talk by M. Gavrilovich about categorical characterizations of basic notions of general topology.

Recall the notion of orthogonality in a category, and that partial orders can be viewed as topological spaces (Alexandrov topology).

• A space $X$ is connected iff $X \to \{\bullet\}$ is left orthogonal to $\{\bullet,\bullet\} \to \{\bullet\}$
• A map $f : X \to Y$ is injective iff it is right orthogonal to $\{\bullet,\bullet\} \to \{\bullet\}$
• A map $f : X \to Y$ is surjective iff it is right orthogonal to $\emptyset \to \{\bullet\}$
• A space $X$ is Hausdorff iff every injective map $\{\bullet,\bullet\} \to X$ is left orthogonal to $\{a < b > a'\} \to \{\bullet\}$
• A space $X$ is compact iff $\emptyset \to X$ is left orthogonal to $\coprod_{\beta<\alpha} \beta \to \alpha$ for every limit ordinal $\alpha$.
• As you say, $X$ is normal iff every closed embedding $A \hookrightarrow X$ is left orthogonal to $\mathbb{R} \to \{\bullet\}$
• Very interesting! But do you mean "lifting property" instead of "orthogonal" in some of these? Jul 2, 2013 at 10:59
• Its interesting that they're all expressed in terms of the same concept - orthogonality. Nlab has an alternative characterisation for $X$ a connected space - when its representable functor $hom[X,-]$ preserves coproducts. Jul 2, 2013 at 11:13
• Presumably you mean that you attended a talk? Jul 2, 2013 at 11:33
• Where can I find proofs of these characterizations?
– user153312
Sep 14, 2015 at 8:50
• I don't know, but the proofs are trivial and just use the definitions. Perhaps you may want to have a look at arxiv.org/pdf/1408.6710.pdf, the paper by M. Gavrilovich about this. Sep 14, 2015 at 14:24