Thread: Interior of a product space

1. Interior of a product space

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.

2. Re: Interior of a product space

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.