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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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Hi x and y, you seem to be new here. For some basic information about writing math at this site see e.g. here, here, here and here. You may find that better formatted questions lead to more responses! –  Tom Oldfield Dec 19 '12 at 18:25
    
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? –  William Dec 19 '12 at 21:43

1 Answer 1

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

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

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