@DavidRoberts Yeah I agree. I couldn't see any way that the Tietze Extension Theorem could be of any use outside of the context of normal spaces, so it seemed like a very strange comment.

Ah thanks! So it seems then that ANY single non-$T_0$-space, according to that reference, is a cogenerator for $Top$?

Oh wow there's a lot written about this Sierpinski space, haha. Okay... so I think I want to say it's a cogenerator of $T_0$-spaces?

@DavidRoberts ah ok, so in other words, the Sierpinski space cannot be a cogenerator for all of $Top$?

Yeah I don't know HHR very well, but generally to any nice enough subcategory there are associated two functors: localization and colocalization, and they are related. Y a fiber sequence. I suspect that is what's happening here.

Yeah I am pretty sure this is exactly controlled by the convergence of the EMSS. I think Brooke Shipley has a nice paper about this, IIRC.

Tyler and @DylanWilson, it's seems intuitively obvious, but does it immediately follow from the above argument (Tyler's not Dylan's) that the filtration is of the form $A\to M\to\ldots$? Clearly the bottom level of the filtration itself is $A$ (since the first filtration quotient is $A$). But does the fact that the next filtration quotient is equivalent to $M/A$ imply that the filtration itself must start with $A$ and $M$?

@dylanwilson yeah that's right, that's a mistake. Or in other words, A and M are the same up through some degree.

Wow Piotr that's awesome! Thanks! This is really similar to the by which McGibbon studies phantom maps in the article I cited.