Understanding Urysohn's Lemma: Explaining the "If" Part

[ehrenfest]

Homework Statement

Urysohn's lemma

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

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

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]

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]

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]

I guess that makes sense, since balls around around 0 can have no negative numbers in them, so they are really just half-balls.