Consider the following interesting theorem (7.5.7, p.294 in *Topology and Groupoids* by Ronald Brown):

**Gluing theorem for adjunction spaces:**
*Suppose that we have the following commutative diagram of topological spaces and continuous maps:*

*where* $\varphi_{A}$, $\varphi_{X}$ *and* $\varphi_{Y}$ *are homotopy equivalences, and the inclusions* $i$ *and* $i'$ *are closed cofibrations. Then the map*

$$ \varphi:X\cup_{f}Y\rightarrow X'\cup_{f\phantom{l}'}Y' $$

*induced by* $\varphi_{A}$, $\varphi_{X}$ *and* $\varphi_{Y}$ *is a homotopy equivalence*.

In this post I asked a question that would become a corollary if the gluing theorem were true for arbitrary cofibrations $i$ and $i'$. **I would like to know if this is the case. Otherwise, does anybody know of a counterexample?**

Perhaps, it is possible that although the map $\varphi$ may not in general be a homotopy equivalence when the cofibrations $i$ and $i'$ are arbitrary, the adjunction spaces $Y\cup_{f}X$ and $Y'\cup_{f'}X'$ will nonetheless be homotopy equivalent. A possible approach to this would be to try to find a third space containing both as deformation retracts.

**Note on notation:** That an inclusion map $j:B\rightarrow Z$ is a closed cofibration means that it is a cofibration (i.e., the pair $(B,Z)$ has the homotopy extension property) and the set $B$ is closed in $X'$.

Thank you