why the open interval (0,1) in R is not compact

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

why the open interval (0,1) in R is not compact

Postby Hom » Tue Oct 04, 2011 12:23 am

Some book is using An=(1/n,1-1/n) as the open subsets. It reaches a conclusion that is not finite.

I am wondering can we just use open subsets like (0,1) itself or (0,0.5)U(0.5,1)to cover (0,1)? what's wrong with this idea?

THanks,

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

Re: why the open interval (0,1) in R is not compact

Postby owlpride » Tue Oct 04, 2011 3:52 am

Back up a second. What's your definition of compactness? Judging from your approach, I assume it's "a space is compact if any open cover has a finite subcover"?

Yes, you can cover (0,1) by itself. Or by (0,0.55) union (0.45,1). Both are finite open covers, but that does not tell you anything about the compactness of (0,1).

What you need to be worried about are infinite open covers. If a space is compact, any and every infinite open cover admits a finite subcover. So let's look at the open cover An=(1/n,1-1/n). Does it admit a finite subcover? If not, (0,1) is not compact.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: why the open interval (0,1) in R is not compact

Postby Hom » Fri Oct 07, 2011 11:03 pm

owlpride wrote:Back up a second. What's your definition of compactness? Judging from your approach, I assume it's "a space is compact if any open cover has a finite subcover"?

Yes, you can cover (0,1) by itself. Or by (0,0.55) union (0.45,1). Both are finite open covers, but that does not tell you anything about the compactness of (0,1).

What you need to be worried about are infinite open covers. If a space is compact, any and every infinite open cover admits a finite subcover. So let's look at the open cover An=(1/n,1-1/n). Does it admit a finite subcover? If not, (0,1) is not compact.


Oh. right. I see. Every open covering has to contain a finite sub collection. This counter example will fail it.

Thank you very much!!

aleph naught
Posts: 1
Joined: Thu Oct 13, 2011 7:48 pm

Re: why the open interval (0,1) in R is not compact

Postby aleph naught » Thu Oct 13, 2011 8:04 pm

If you're doing the Math GRE you'll definitely want to take the Heine Borel theorem for granted:

"A subset of a metric space is compact if and only if it is closed and bounded."

Or for Euclidean space:

"A subset of a R^n is compact if and only if it is closed and bounded."

The only if direction essentially can be proven with the same trick you use to provide a counterexample for (0,1).

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: why the open interval (0,1) in R is not compact

Postby blitzer6266 » Thu Oct 13, 2011 9:16 pm

Your Heine-Borel statement for metric spaces is not correct. You need completeness and total boundedness




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