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Math Help - topology- compactness, interior=empty

  1. #1
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    topology- compactness, interior=empty

    I need help with this exercise..

    Let \{ X_\alpha\}_{\alpha \in I} be a collection of topological spaces. X=\prod_{\alpha \in I}X_\alpha
    1)prove: If an infinite number of  X_\alpha are non-compact, then any compact subset in \prod_{\alpha \in I}X_\alpha has empty interior. 2)is there a need for every compact set  K\subseteq Xto be dense nowhere in  X( Int \overset{-}{K}=\emptyset )?

    ..I look at the product \prod_{n \in \mathbb N}X_n where all X_n are equal to set \mathbb R \cup \{c\}  (c \notin \mathbb R) with a topology whose basis consists of sets in form of \langle a,b \rangle \cup \{c\}, a,b\in\mathbb R

    Thank you!
    Last edited by tom007; May 4th 2011 at 11:49 PM.
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  2. #2
    Senior Member Tinyboss's Avatar
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    Basis elements in the product topology are the entire space in all but finitely many factors. So if infinitely many factor spaces are non-compact, then no basis element is contained in a compact set.
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  3. #3
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    mm..cool
    and that means that it has empty interior..tnx
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