To show that product $Z=X×Y$ in the product topology is a CW complex

Question

I would like to prove the following:

If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here)

In order to prove it, it needs to satisfy 3 conditions as in the definition of CW complex:

I have proven the condition 1 and 2, they are quite straight forward and even hold without assumption that $Y$ being locally compact. I got stuck to prove the third condition, and I think it is where the locally compactness is used. So, suppose $A\cap \bar{e_\alpha}\times \bar{e_\beta}$ is closed in $\bar{e_\alpha}\times \bar{e_\beta},\forall\alpha,\beta$. How to show that $A$ is closed in $X\times Y$?

Any help is appreciated :)

Answer 1

I think this is related (or maybe follows from) the result by E. Michael that the product of a locally compact space and a $k$-space is again a $k$-space.

Answer 2

This follows from the standard result given in, for example, Topology and Groupoids:

$4.3.2$ Let $f : X \to Y$ be an identification map and let $B$ be locally compact. Then
$$f × 1 : X × B \to Y × B$$ is an identification map.

The definition of locally compact here is that each point has a base of compact neighbourhoods.

This inconvenient restriction led to the notion of convenient category of topological spaces. See also Section 5.9 of Topology and Groupoids.