Suppose we have a pushout of pointed spaces, where $i$ and $j$ are cofibrations.
Let $M$ and $N$ be the cofibres of $i$ and $j$ respectively. Then show there is a homeomorphism $M\cong N$.
This question is quite short, and there are only a few definitions involved. I know the definitions of pushout, cofibre and cofibration, but I can't see how to show this. My guess is that I'm missing (or overlooking) some key result.
Any help is greatly appreciated! Thanks in advance
Asking for a homeomorphism in this case seems too strong, because to me the (homotopy) cofiber is only well defined up to homotopy. There are point-set models of cofibers, e.g., the (reduced) mapping cone, but I wouldn't expect properties that aren't homotopy invariant to be preserved.
Concretely, let's assume that by cofiber you mean the reduced mapping cone construction. Consider the pushout square $$\begin{array}{ccc} * & \xrightarrow{\mathrm{id}} & * \\ \downarrow & & \downarrow \\ ([0,1],0) & \xrightarrow{\mathrm{id}} & ([0,1],0). \end{array}$$
The cofiber of the top row is again the point $M = *$, but the cofiber of the bottom row is homeomorphic to the 2-simplex, i.e., $N \cong \Delta^2$. These spaces are obviously homotopy equivalent, but they are not homeomorphic: their cardinalities are different, so there isn't even a bijection between them.