# Is the product of two good pairs, itself a good pair?

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!