Thread: Interior of a product space

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

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

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

Originally Posted by FGT12

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

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

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

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

Put those ideas together.