Chapter 2.19, Problem 1E

Prove Theorem $19.2$.

Theorem $19.2.$ Suppose the topology on each space $X_{\alpha }$ is given by a basis $B_{\alpha }$. The collection of all sets of the form

${\displaystyle \underset{\alpha \in J}{\prod }{B}_{\alpha }}$,

where $B_{\alpha }\in B_{\alpha }$ for each $\alpha$, will serve as a basis for the box topology on ${\displaystyle {\prod }_{\alpha \in J}{X}_{\alpha }}$.

The collection of all sets of the same form, where $B_{\alpha }\in B_{\alpha }$ for finitely many indices $\alpha$ and $B_{\alpha }=X_{\alpha }$ for all the remaining indices, will serve as a basis for the product topology ${\displaystyle {\prod }_{\alpha \in J}{X}_{\alpha }}$.