I asked [this](https://math.stackexchange.com/questions/1190984/weak-equivalence-testable-on-invariant-open-covers) question on math.stackexchange, but did not get an answer.


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 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 $f\colon X\rightarrow X'$.
> 
> 
> 


1. 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?
2. What about the corresponding claim with homotopy equivalences instead of weak equivalences?



When are (weak) homotopy equivalence testable on open covers?
For weak homotopy equivalences this holds always (Theorem 6.7.9 in tom Dieck's *Algebraic Topology*).


For homotopy equivalences this holds provided the open covers are numerable (Theorem 4.2.7 loc. cit.)


