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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!

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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.

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