Urysohn's lemma


by ehrenfest
Tags: lemma, urysohn
ehrenfest
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#1
Oct5-07, 12:08 AM
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1. The problem statement, all variables and given/known data
Urysohn's lemma

My book says that the "if" part of Urysohn's lemma is obvious with no explanation. Can someone explain why?



2. Relevant equations



3. The attempt at a solution
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HallsofIvy
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Oct5-07, 06:14 AM
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It would have been a good idea to actually state Urysohn's lemma as it is given in your book. Sometimes statements vary from one book to another. In particular you should note that Urysohn's lemma only applies in NORMAL spaces. What is the definition of a "Normal" topological space?
ehrenfest
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Oct5-07, 11:53 PM
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Quote Quote by HallsofIvy View Post
It would have been a good idea to actually state Urysohn's lemma as it is given in your book. Sometimes statements vary from one book to another. In particular you should note that Urysohn's lemma only applies in NORMAL spaces. What is the definition of a "Normal" topological space?
Sorry. I meant to put a link to Wikipedia, which has the same statement of Urysohn's Lemma as that in my book.

http://en.wikipedia.org/wiki/Urysohns_lemma

It comes down to whether the sets [0,1/2) and (1/2,1] are open. Apparently this is obvious to other people, but it seems counterintuitive to me because I thought open sets were open intervals.

Hurkyl
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Oct5-07, 11:56 PM
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Urysohn's lemma


Quote Quote by ehrenfest View Post
Sorry. I meant to put a link to Wikipedia, which has the same statement of Urysohn's Lemma as that in my book.

http://en.wikipedia.org/wiki/Urysohns_lemma

It comes down to whether the sets [0,1/2) and (1/2,1] are open. Apparently this is obvious to other people, but it seems counterintuitive to me because I thought open sets were open intervals.
[0, 1/2) is not an open subset of R
[0, 1/2) is an open subset of [0, 1].
ehrenfest
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#5
Oct6-07, 12:14 AM
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I guess that makes sense, since balls around around 0 can have no negative numbers in them, so they are really just half-balls.


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