Chapter 2.16, Problem 5E

Let $X$ and $X^{\prime }$ denote a single set in the topologies $\tau$ and ${\tau }^{\prime }$, respectively; let $Y$ and $Y^{\prime }$ denote a single set in the topologies $\mu$ and ${\mu }^{\prime }$, respectively. Assume these sets are nonempty.

(a) Show that if ${\tau }^{\prime }\supset \tau$ and $u^{\prime }\supset u$, then the product topology on $X^{\prime }\times Y^{\prime }$ is finer than the product topology on $X\times Y$.

(b) Does the converse of (a) hold? Justify your answer.