Product Topologies
Definition
Given a family of topological spaces
$\mathcal{F} = \set{(X_\alpha ,
\mathcal{T} _\alpha }_{\alpha \in I}$, the
product topology of
$\mathcal{F} $ is the topology on $\prod_{\alpha
\in I} X_\alpha $ generated by the set
\[
\Set*{ p_{\beta }^{-1}(U_\beta )) }{ (\exists \beta )(U_\beta \in
X_{\beta }) }
\]