Products of well-pointed spaces

Question

Suppose $(X,x_0)$ and $(Y,y_0)$ are well-pointed topological spaces, i.e. the inclusions of the respective base points are cofibrations. Does it follow that $(X \times Y, (x_0,y_0))$ is well-pointed as well? I thought that this should be true, but I couldn't figure out a simple argument.

Answer 1

I'm not sure whether the comment is entirely correct. My textbook (tom Dieck) says that $X\times A\to Y\times X$ is a cofibration if $X$ is closed in $Y$ or $A$ is locally compact. As points are compact, you would just have to assume that one base point is closed in its ambient space. Then you could use that compositions of cofibrations are cofibrations as stated in the comment.