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 | | From: | Martin | | Subject: | Hausdorff Theorem | | Date: | Sun, 23 Jan 2005 23:53:12 +0000 (UTC) |
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Dear all,
I'm trying to find a proof of the following Hausdorff theorem:
If f is a function on [a,b] and C(f),the set of points of continuity of f, is dense in [a,b], then C(f) is not a set of first category
in [a,b].
As the only reference I have the highly inaccessible Hausdorff's
"Grundz"uge der Mengenlehre" from 1914. I will be most grateful
for any help towards the proof of this important theorem.
Best regards,
Martin
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| | From: | William Elliot | | Subject: | Re: Hausdorff Theorem | | Date: | Mon, 24 Jan 2005 01:13:10 -0800 |
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| From: Martin
Newsgroups: sci.math
Subject: Hausdorff Theorem
> If f is a function on [a,b] and C(f),
> the set of points of continuity of f, is dense in [a,b],
> then C(f) is not a set of first category in [a,b].
C = C(f) is G-delta by theorem for metric spaces, ie
for all n in N, some open U_n with C = /\_n U_n.
Thus
R\C = \/_n R\U_n
int cl R\U_n = int R\U_n subset int R\C = R\cl C = nulset
making R\C sparse, ie 1st category.
So C can't be sparse, otherwise R would be sparse
and you'd be caught by the big Baire.
In general for any topological space X and metric space Y
If f:X -> Y and Y isn't sparse, for example a Baire space,
and if C(f) is dense, then C(f) isn't meager.
Be patient and polite, wait for answer from sci.math,
before rushing off to sci.math.research. Are you a researcher?
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