Interior of a product space

**FGT12** · September 16th 2011, 01:19 PM

Suppose that $X,Y$ are spaces and that $A \subseteq X, B \subseteq Y$ . Prove that the interior of $A \times B$ is $int(A) \times int (B)$ .

$int(A \times B) = \bigcup_{ U \subseteq A \times B} U = \bigcup_{ U \subseteq A} U \times \bigcup_{ U \subseteq B} U$ where $U$ is open

Can I justify the last equality?? Im not sure.

**Plato** · September 16th 2011, 02:22 PM

Originally Posted by FGT12

Suppose that $X,Y$ are spaces and that $A \subseteq X, B \subseteq Y$ . Prove that the interior of $A \times B$ is $int(A) \times int (B)$ .

$int(A \times B) = \bigcup_{ U \subseteq A \times B} U = \bigcup_{ U \subseteq A} U \times \bigcup_{ U \subseteq B} U$ where $U$ is open

Notation: let $A^o$ denote the interior of $A$
What does it mean for $s\in A^o~\&~t\in B^o~?$

How are open set defined is $X\times Y~?$

Put those ideas together.

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