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Math Help - Interior of a product space

  1. #1
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    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.
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  2. #2
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    Re: Interior of a product space

    Quote Originally Posted by FGT12 View Post
    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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