Theorem 25.1:

Let $X$ and $B$ be topological spaces, $A$ be a closed subspace of $X$ and $f:A\longrightarrow B$ be a continuous map. Then the space $X\sqcup_f B$ is the push-out for the following diagram

$\begin{CD} A @> i >> X \\ @V{f}VV @. \\ B @. \\ \end{CD}$

where $i:A\longrightarrow X$ denotes the inclusion map.