Does geometric realization commute with passing to the compactly generated topology?

My question is in the title, but here is a more detailed formulation:

Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated spaces, as defined in Strickland's notes. We have a functor $k: Top \rightarrow CGTop$ that replaces the topology on a space by the associated compactly generated topology.

If X is a simplicial topological space (a simplicial object in the category of all topological spaces) is the identity map $|kX|\rightarrow k|X|$ a homeomorphism?

Here $|-|$ denotes geometric realization. I'm actually most interested in the "thick" realization (where the defining equivalence relation uses only the face maps), but the question makes sense for both the thick and thin versions.

It follows from the results in the above notes that $|kX|$ is compactly generated (being a quotient of a disjoint union of compactly generated spaces), and that the identity map $|kX|\rightarrow k|X|$ is continuous.

Here is a more general question: if $X$ is a topological space and $\sim$ is an equivalence relation on $X$, then $(kX)/{\sim}$ is compactly generated and the identity map $(kX)/{\sim} \to k(X/{\sim})$ is continuous. Is this map always a homeomorphism?

• Your second question can be rephrased as asking whether the topology of kR is a subspace of k(X×X), where R < X×X is the equivalence relation with the subspace topology. I don't know if it makes it easier or harder... – Denis Nardin Jun 2 '18 at 19:44
• @DenisNardin My guess is that the second question is false in general, and I can imagine that your formulation might be the best way to look for a counterexample. – Dan Ramras Jun 2 '18 at 20:42
• @DanRamras Should the equivalence relation $\sim$ be closed in $X\times X$ or not? – Taras Banakh Jun 6 '18 at 11:20
• @TarasBanakh I suspect the following is true, but haven't checked carefully: if the spaces $X_n$ are all Hausdorff, then the equivalence relation that defines the "thick" geometric realization is closed. Thus a positive answer to the my "general question" for closed equivalence relations ought to give a positive answer to the main question if one uses thick realization and assumes the spaces are Hausdorff. I'd be happy if that could be proven. – Dan Ramras Jun 8 '18 at 2:41
• For what it's worth, I now think a better question is whether the maps in my question are weak homotopy equivalences, rather than homeomorphisms. – Dan Ramras Jul 11 '18 at 3:15