$f\times\text{id}$ is closed for a closed map $f$ with $f^{-1}(y)$ compact $\forall y\in Y$

Question

Let $f:X\rightarrow Y$ be a map between topological spaces such that $f$ is closed and $f^{-1}(y)$ is compact in $X$ for every $y\in Y$.

Now I have to show that the map $f\times\text{id}:X\times T \rightarrow Y\times T$ is closed for every topological space $T$.

I don't really know how to start here. Therefore any hint or advice is appreciated!

Answer 1

You can find this very statement in page 90 of the book Topology and Groupoids by R. Brown.

The proof basically show that the complement of each $(f\times{id})(C)$ for $C$ closed in $X\times T$ is open in $Y\times T$ and relies on the following "tube lemma" (also shown in the book).

Lemma. Let $X,Y$ topological spaces and let $A\subset X,B\subset Y$ be two compact subsets. Let furthermore $W$ be an open subset of $X\times Y$ containing $A\times B$. Then there are open sets $U\supset A, V\supset B$ such that $A\times B\subset U\times V\subset W$.