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Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.

There is the CG-ification $X_{CG}$ of a space $X$ and the WH-ification $X_{WH}$ of a compactly generated space $X$. These provide adjoint pairs between $Top$ and $CG$ and also between $CG$ and $CGWH$. Combined, they give me the CGWH-fication $(X_{CG})_{WH}=X_{CGWH}$ of a space $X$. The latter one is the one used implicitly in most of today's algebraic topology.

I'm interested in the question which of these constructions can change the (weak) homotopy type and which of them preserves it. It seems to me that at least in many cases, the weak homotopy type must be preserved. I really don't want to have $\Omega X$ to have a different homotopy type depending on whether I used the compactly generated topology on the mapping space or not.

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  • $\begingroup$ Are you sure that the weakHausdorffication works for every space? I know that it can be applied to a $k$-space $X$, where it is $X/R$ with $R$ being the smallest closed equivalence relation in the $k$-product $X\times X$. This yields a functor $CG\to CGWH$. $\endgroup$ – Stefan Hamcke Jun 17 '15 at 13:48
  • $\begingroup$ You are right. I will edit the question. $\endgroup$ – Klaus Jun 17 '15 at 13:50
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It is very easy to see that CG-ification preserves weak homotopy type: it is a right adjoint, so for any CG space $K$, the maps $K\to X_{CG}$ are the same as the maps $K\to X$. Letting $K$ be $S^n$ and $S^n\times I$ immediately gives that the map $X_{CG}\to X$ is a weak equivalence.

However, WH-ification does not preserve weak homotopy type. For instance, if $X$ is a space with finitely many points, then $X$ is CG, and $X_{WH}$ will always be discrete (since it is a finite $T_1$ space). However, finite spaces can have the weak homotopy type of any finite CW-complex! So WH-ification can really destroy a lot of homotopy-theoretic information when applied to arbitrary spaces. This rarely is a cause for concern though. For instance, you mentioned being worried about the topology on loopspaces, but the CG-ification of the loopspace of a CGWH space is automatically already WH (see Proposition 2.24 of this excellent paper, for instance).

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  • $\begingroup$ "However, finite spaces can have the weak homotopy type of any finite CW-complex!" This is impressive. Do you have a reference for this fact? $\endgroup$ – Klaus Jun 17 '15 at 14:19
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    $\begingroup$ @Klaus: It's a theorem of McCord; here's the original reference. One version of the statement is as follows. Let $X$ be a simplicial complex, and let $Y$ be the quotient of $X$ obtained by collapsing the interior of each simplex of $X$ to a point. Then the quotient map $X\to Y$ is a weak equivalence. When $X$ is a finite simplicial complex, of course, this $Y$ is a finite space. $\endgroup$ – Eric Wofsey Jun 17 '15 at 15:21

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