This question begins with a sort of mysterious comment at the bottom of [this Wikipedia page](https://en.wikipedia.org/wiki/Injective_cogenerator#In_general_topology) on injective cogenerators. There, it is said, without citation or proof, that as a result of the [Tietze Extension Theorem](https://en.wikipedia.org/wiki/Tietze_extension_theorem), the interval $I=[0,1]$ is an injective cogenerator for categories of topological spaces satisfying [separation axioms](https://en.wikipedia.org/wiki/Separation_axiom) (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](https://ncatlab.org/nlab/show/cogenerator) 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. 



Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
Perhaps I should have been a little better at Googling before posting this question, but it seems to be answered, to a degree, in a paper from 1980 by Giuli, cited below. In particular, any epi-reflective subcategory, i.e. one that is closed under products and subspaces (hence whose inclusion is limit preserving), has a system of cogenerators that do not form a set. This property is referred to by Giuli as being "weakly initial." This, if I'm reading this right, looks like a system of spaces $X(\Lambda)$ for every cardinal $\Lambda$. Now, in some nice cases, e.g. $T\_0$, $T\_4$, Tychonoff, and compact Hausdorff, there's in fact a *single* cogenerator. But this is not the case (according to that paper, though not proven) for $T\_1$, $T\_2$, $T\_3$ or Urysohn spaces. In fact, Giuli says that there are not even systems of cogenerators *known* for $T\_2$, $T\_3$, and Urysohn (or $T\_{2\frac{1}{2}})$.


However, Giuli does confirm that the interval $[0,1]$ is a cogenerator for Tychonoff spaces and that the Sierpinski space is a cogenerator for $T\_0$-spaces (apparently). All of this discussion begins around Theorem 1.2 of that paper. 


*Giuli, Eraldo*, [**Bases of topological epi-reflections**](http://dx.doi.org/10.1016/0166-8641(80)90025-5), Topology Appl. 11, 265-273 (1980). [ZBL0441.18012](https://zbmath.org/?q=an:0441.18012).


