A subset $S$ of a topological space $X$ is called semi-open if there exists an open set $O$ such that $O \subset S \subset \mbox{cl}O$ where by $\mbox{cl}O$ I mean the closure of the set $O$.
We can define semi-open equivalently: A subset $S$ of a topological space $X$ is called semi-open if $S\subset \mbox{cl}(\mbox{int}S)$. Where $\mbox{int}S$ is the interior of the set $S$.
My question is
If $\{A_i\}$ is a locally finite family of semi-open sets in a topological space $X$ and if $\{B_i\}$ is a locally finite family of semi-open sets in a topological space $Y$
Is $\{A_i \times B_i\}$ a locally finite family of semi-open sets in the product space $X\times Y$?