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Prove or disprove: $(A_i)_{i=1}^\infty$ are closed subsets in a complete metric space. Assume that there is an open ball in the $\bigcup\limits_{i=1}^\infty A_i$ , so exists $k$ s.t $A_k$ contains an open ball as well.

Thank you!

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Please elucidate: What did you try, what are your thoughts, where are you stuck? –  Tim van Beek Oct 25 '11 at 15:35
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is it a homework? –  Ilya Oct 25 '11 at 15:39
    
it is homework, I couldn't find any example to disprove that but I have no clue how to start the proof, if it is true, so I will be glad to get a hint or direction... –  user18217 Oct 25 '11 at 15:54

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up vote 8 down vote accepted

Hint: Baire category theorem. If no $A_k$ contains an open ball, then all $A_k$'s are nowhere dense, hence $\bigcup_{k} A_k$ ...

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thanks! I will try this –  user18217 Oct 25 '11 at 16:13

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