I have been struggling with the following problem.
Consider the pushout for topological spaces (or adjunction space) $B \cup_A C$ obtained by gluing together $B$ and $C$ along $A$ by means of continuous maps $f$ and $g$. $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} A & \ra{f} & B\\ \da{g} & & \da{g'}\\ C & \ras{f'} & B \cup_A C\\ \end{array} $ show that if $f$ is open and injective then $f'$ (the pushout of $f$) is open.
I have tried to take an open set $U \subseteq C$, and show that $f'(U)$ is open, for this I have to show that $g'^{-1}(f'(U))$ is open in $B$, so I tried it to compare it somehow to $f(g^{-1}(U))$ which is an open in $B$, but have not been able to make any progress after that.
Also I'm not sure how to use the injectivity of $f$, is it maybe for its inverse? so I can take the equality $f'(g(x)) = g'(f(x))$ and maybe manipulate it somehow to get something like $f'(g(f^-1(x)) = g'(x), x \in B$, but I'm not sure if that is helpful.
Any help would be appreciated.