$CW$ topology on $X \times Y$ coincides with the product topology I think I heard somewhere that for $X,Y$ (both $CW$ complex) endowing the $CW$ complex $X \times Y$ with its weak topology is equivalent to endowing $X \times Y$ with the usual product topology.
Is this result true?
It's certainly true when $Y=[0,1]$, often use in proofs involving continuity of homotopies from $CW$ complex. I'd be interested in references or proofs, any help would be appreciated, thanks in andvance.
 A: $\textbf{Theorem:}$ Let $X$ and $Y$ be CW complexes. Then $X \times Y$ is a CW complex if and only if one of the following holds:
$(1)$ Either $X$ or $Y$ is locally finite.
$(2)$ Either $X$ or $Y$ has countably many cells in each connected component, and the other has fewer than $\mathfrak b$ many cells in each connected
component.
$\bullet$ For a proof  of the Theorem  and the definitions of the locally finite, the cardinal $\mathfrak b($called the bounding number, and $\aleph_1\leq \mathfrak b\leq 2^{\aleph_0}$$)$ see $\text{Theorem 1}$, and $\text{Definition 7}$, $\text{Definition 3}$; respectively of this paper PRODUCTS OF CW COMPLEXES by
ANDREW D. BROOKE-TAYLOR.

$\bullet$ Now, I will quote some portions of the book The Topology of CW Complexes by A.T. Lundell, S. Weingram.
Let $X_1, X_2$ be two CW-complexes and $X_1 \otimes X_2$ denote the space for which the underlying set is the product $X_1 \times X_2$ and the topology is the weak topology with respect to the product cells. Then $X_1 \otimes X_2$ is a CW complex.
$\textbf{Proposition 5.1.}$ The identity map $\text{Id}\colon X_1\otimes X_2\to Χ_1\times X_2$ is a continuous bijection. Thus the spaces $X_1\otimes X_2$ and $X_1 \times X_2$ differ only in that the CW topology may be finer than the product topology.
$\textbf{Theorem 5.2.}$ If at least one of the two CW-complexes has only countably many cells, or one of the two CW-complexes has finitely many cells, then $\text{Id}\colon X_1\otimes X_2\to Χ_1\times X_2$ is a homeomorphism. Thus the CW topology and the product topology on the product set $X_1 \times  X_2$ coincide in this case.
$\textbf{Corollary 5.3.}$ A subset $K \subseteq X \otimes Y$ is compact if and only if it is compact in $X \times Y$.

$\textbf{Fact:}$ Let $X_1,X_2$ be two CW-complexes, then $\text{Id}\colon X_1\otimes X_2\to Χ_1\times X_2$ is a CW-approximation, i.e., given any $(x, y) \in X \times Y$ the induced map $\pi_n(\text{Id})\colon\pi_n \big(X \otimes Y , (x, y)\big) \to \pi_n \big(X \times Y, (x, y)\big)$ is an isomorphism for all $n\geq 1$.
To prove this fact, one needs to use the above $\text{Theorem 5.2.}$ and $\text{Corollary 5.3.}$
