This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze Extension Theorem, the interval $I=[0,1]$ is an injective cogenerator for categories of topological spaces satisfying separation axioms (e.g. Hausdorff, Tychonoff, Kolmogorov, or other of the various $T_{i}$ conditions).

So there are a few questions here:

(0) This question is basically terminological. The Wikipedia page says that an injective cogenerator is simply an object that admits a non-zero (although in a general category I'm not sure what a zero map would be anyway) map from every non-zero object. But the nlab page indicates that we should rather define a cogenerator to be an object whose represented functor is faithful, so that's the definition I'm working with for now. Are these equivalent in the case that "zero map" has a meaning?

(1) Is it true that the interval is an injective cogenerator for any of the categories of spaces subject to separation axioms? If so, can you give an idea of how the proof would go? It seems likely that it would work for Tychonoff spaces at least?

(2) If not the interval, do the other categories of spaces subject to separation axioms admit *any* (set of) cogenerators?

(3) Does the full category of topological spaces have a cogenerator? I was thinking that it might be the two point space $X=\{a,b\}$ with topology given by $\{\{a,b\},\{b\},\emptyset\}$, but I might be wrong here.

soberspaces. There are non-sober $T_0$ spaces, I'm pretty sure. $\endgroup$ – David Roberts May 24 '18 at 3:24