Urysohn's Lemma section in Munkres

Forum for the GRE subject test in mathematics.
dasgut
Posts: 250
Joined: Thu Jan 21, 2010 12:20 am

Urysohn's Lemma section in Munkres

Postby dasgut » Fri Jan 24, 2014 9:35 am

I'm going over Munkres's Topology again. Is it just me or are the problems in the Urysohn's Lemma section exceptionally hard? I've gotten less than half on my own while I've been able to get 90% of the other problems otherwise!

l0019
Posts: 14
Joined: Sat Dec 07, 2013 4:31 pm

Re: Urysohn's Lemma section in Munkres

Postby l0019 » Fri Jan 24, 2014 9:58 am

I remember the problems getting much more difficult at a certain point in that text, I think it may have been around Urysohn's lemma. I think the problem where you have to prove that R^J is not normal when J is uncountable was tricky for me

dasgut
Posts: 250
Joined: Thu Jan 21, 2010 12:20 am

Re: Urysohn's Lemma section in Munkres

Postby dasgut » Fri Jan 24, 2014 10:25 am

Ha. I actually thought that was the easy problem in the chapter! For myself, the questions on G delta sets and perfect normality were particularly painful.

l0019
Posts: 14
Joined: Sat Dec 07, 2013 4:31 pm

Re: Urysohn's Lemma section in Munkres

Postby l0019 » Fri Jan 24, 2014 11:00 am

Interesting. I can't remember what I thought was tricky for me about the R^J problem, it is one where they walk you through the proof step by step, right? I think I was confused how he wanted you to do one of the steps. What is the G delta sets problem? Been a year or so since I've looked at that book

dasgut
Posts: 250
Joined: Thu Jan 21, 2010 12:20 am

Re: Urysohn's Lemma section in Munkres

Postby dasgut » Fri Jan 24, 2014 11:15 am

l0019 wrote:Interesting. I can't remember what I thought was tricky for me about the R^J problem, it is one where they walk you through the proof step by step, right? I think I was confused how he wanted you to do one of the steps. What is the G delta sets problem? Been a year or so since I've looked at that book


For the 1st G delta step problem they want you to show that, in a normal space, for a closed set C there exists a continuous, real-valued function f s.t. f(C)={0} and f(x)>0 when x doesn't belong to C iff C is G delta.

The second one wants you to show that for disjoint closed sets C and D in normal space there exists a continuous, real-valued function f s.t. f(C)={0}, f(D)=1, and 1>f(x)>0 when x doesn't belong to C or D iff C and D are G delta.

The => direction is obviously pretty easy, but the <= is damn hard in my opinion. I had to look online.

Basically, for the first problem, you follow the proof of Urysohn's Lemma but you include an extra trick to control how you build the sets about C that define f. I don't think I could have come up with such a trick on my own without hints! I'm pretty sure you can make a similar argument for the second one, but there's a really easy trick to get by that complex construction. Using the "metric like" properties of the first problem's sorta, kinda pseudo-premetric function you can build the necessary function for sets C and D.




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