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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 $X\times Y$?

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    Correct me if I'm being too naive, but it seems that the question is trivial. Indeed, local finiteness is trivial straight from definition. Now assuming that $X \times Y$ is taken with product topology, semi-openess should also be trivial from definition, since open sets in $X\times Y$ are of the form $U\times V$ where $U$ and $V$ are open in $X$ and $Y$, respectively. What is the catch?2012-12-19

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This is quite trivial: for product spaces $X \times Y$ we have the following well-known identities:

$\mbox{int}\,(A \times B) = \mbox{int}\,A \times \mbox{int}\,B$

$\mbox{cl}\,(A \times B) = \mbox{cl}\,A \times \mbox{cl}\,B$

for $A \subset X, B \subset Y$.

From this it follows (using your second formulation of semi-open) that the product of a semi-open subset of $X$ with a semi-open subset of $Y$ is semi-open in the $X \times Y$.

Also, independently of this, if $(A_i)_{i \in I}$ is a locally finite family of subsets of $X$ and $(B_i)_{i \in I}$ is a locally finite family of subsets of $Y$, then $(A_i \times B_i)_{i \in I}$ is a locally finite family of subsets of $X \times Y$: for $(x,y)$ in $X \times Y$, let $O$ be a neighbourhood of $x$ in $X$ that intersects at most finitely $A_i$, and similarly let $O'$ be a neighbourhood of $y$ in $Y$ that intersects at most finitely many $B_i$, then clearly $O \times O'$ is a neighbourhood of $(x,y)$ that intersects at most finitely many $A_i \times B_i$, as

$ (O \times O') \cap (A_i \times B_i) \neq \emptyset \mbox{ iff } O \cap A_i \neq \emptyset \mbox{ and } O' \cap B_i \neq \emptyset$