Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets
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I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.
The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.
How can I use this on the proof?
general-topology compactness baire-category
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
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add a comment |
$begingroup$
I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.
The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.
How can I use this on the proof?
general-topology compactness baire-category
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
$endgroup$
add a comment |
$begingroup$
I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.
The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.
How can I use this on the proof?
general-topology compactness baire-category
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
$endgroup$
I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.
The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.
How can I use this on the proof?
general-topology compactness baire-category
general-topology compactness baire-category
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
edited Dec 11 '18 at 11:26
José Carlos Santos
162k22130233
162k22130233
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
asked Dec 11 '18 at 10:54
A curious oneA curious one
5318
5318
add a comment |
add a comment |
1 Answer
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Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
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add a comment |
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$begingroup$
Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
add a comment |
$begingroup$
Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
add a comment |
$begingroup$
Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
$endgroup$
Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
answered Dec 11 '18 at 10:57
José Carlos SantosJosé Carlos Santos
162k22130233
162k22130233
add a comment |
add a comment |
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