Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let $X=U\cup V$ and $X'=U'\cup V'$ be an open covers such that $f(U)\subseteq U'$ and $f(V)\subseteq V'$ holds.
Consider the following claim.
If the three restrictions $f\colon U\rightarrow V$, $f\colon V\rightarrow V'$ and $f\colon U\cap U'\rightarrow V\cap V'$ are weak homotopy equivalences, then so is $f\colon X\rightarrow X'$.
- Is this claim true in general? If not, are there mild assumptions on $X$, $X'$ or $X$ and $X'$, such that the claim holds, e.g. does the claim hold if $X$ and $X'$ are Hausdorff spaces?
- What about the corresponding claim with homotopy equivalences instead of weak equivalences?
Not to leave this question open:
Answers can be found at the sequel of this question on mathoverflow .