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I have two problems:

1.- Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$.

2.- Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, then $|X|\leq|P(P(D))|$, where $P(D)$ is the power set of $D$.

If somebody know where I can find the proofs, tell me please.

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    All-caps in questions might be considered a bit rude. Also, is this homework? EDIT: Also also, I'm not sure I see two problems.2011-06-15
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    I only see one question here. What is the other problem?2011-06-15
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    Thanks, @Zev, for editing in the questions.2011-06-15
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    Should the first problem say "...less than or equal to $|X|$" or perhaps "... less than or equal to that of $X$"?2011-06-15

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