This is from Charles Rezk's notes on compactly generated spaces: see here (p.12, proposition 10.6.)
Let $i\colon A\to X, f\colon A\to B$ be maps of compactly generated spaces and let $Y$ be a pushout of $i$ and $f$ in the category of topological spaces. Assume that $i$ is a closed embedding and $f$ is a quotient map. Then $Y$ is compactly generated.
There is a certain step I don't understand. Let $g\colon X\to Y$ and $g'\colon B\to Y$ be maps which go with the pushout $Y$. Rezk claims that $(g\times g)^{-1}(\Delta_Y) = \Delta_X\cup (i\times i)((f\times f)^{-1}(\Delta_B))$ because $i$ is injective but I can't see why $(g\times g)^{-1}(\Delta_Y) \subseteq (i\times i)((f\times f)^{-1}(\Delta_B))$ (here $\Delta$ denoted the diagonal).
I understand that $g$ is also a quotient map because of the pushout properties.
Is there a mistake?
Rezk does not claim there is the inclusion you say.
The pushout is computed as a pushout of sets, so we have a subspace $A$ of $X$ and an equivalence relation on this subspace (given by $f$). Two points of $X$ are glued if and only if they are equal or they are in $A$ and equivalent by the relation. The set $(f×f)^{-1}(_B)$ is the set of couples of points of $A$ we want to glue and we take its image by $i×i$ to know the points in $X$ we glue together.