Want to see whether $(X,A)\times(Y,B)=(X\times Y,A\times Y\cup X\times B)$ is a good pair whenever $(X,A)$ and $(Y,B)$ are good pairs. Searched the text by Hatcher on Algebraic Topology, and couldn't locate so far as I searched. So trying to prove that it is true.
The case relevant to me now is when $A=\{x_0\}\subset X$ and $B=\{y_0\}\subset Y$ are points.
If $x_0$ is a deformation retract of, say $U$ open in $X$ and $y_0$ is a deformation retract of $V$. I see how $U\times Y$ and $X\times V$ deformation-retract to $\{x_0\}\times Y$ and $X\times\{y_0\}$ respectively.
However whether their union $U\times Y\cup X\times V$ deformation retracts to $\{x_0\}\times Y\cup X\times\{y_0\}$ is still a question mark. I tried collapsing $U\times V$ to a point, and then homotoping the resulting space to $\{x_0\}\times Y\cup X\times\{y_0\}$. While $U\times V$ is contractible, the quotient map collapsing $U\times V$ need not be a homotopy equivalence, because I don't know whether the pair $(U\times Y\cup X\times V,U\times V)$ satisfies the homotopy extension property or not. If it does, hurray! But if not, bad news. I am now trying to use obstruction theory, whew!
The answer to the question might turn out to be no which make my efforts futile! Any help is greatly appreciated!
The following is from Topology and Groupoids
7.3.8 Let (X, A) and (Y, B) be closed cofibred pairs. Then the pair (X × Y, A × Y ∪ X × B) is also a closed cofibred pair.
The proof is given there. Chapter 7 of T&G is downloadable from here. The result derives from
Lillig, J. ‘A union theorem for cofibrations’. Arch. Math. 24 (1973) 410–415.
Actually it was a question in the first 1968 edition! You can also find it I think in I M James, Homotopy theory and general topology (Springer) 1984.