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Old 2004-11-05, 01:59   #1
synergy
 
Aug 2004

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Question Topology Question - Help ?!


Does anybody have any suggestions on how to prove that a topological space X is COUNTABLY compact (i.e. every COUNTABLE open cover has a finite subcover), IF AND ONLY IF, EVERY NESTED SEQUENCE of closed nonempty subsets of X has a nonempty intersection?

I also need hints on how to show whether S-omega is SC, CC, LPC, or compact, where SC is sequentially compact(every sequence has a convergent subsequence), CC is countably compact (every COUNTABLE open cover contains a finite subcover, LPC is limit point compact (every infinite subset has a limit point), and compact is EVERY open cover contains a finite subcover). S-omega is the smallest uncountable set (properties: cut it off anywhere before the point omega and you have a countable set, the point omega has no IMMEDIATE predecessor).

I also need to show whether [0,1]^|R, that is, [0,1]x[0,1]x... an uncountable number of times, whether that is sequentially compact, i.e. whether EVERY sequence has a convergent subsequence.

I also need to show that X x Y is limit point compact but not countably compact, where X=the natural numbers, the topology on X is the power set, Y={0,1} and its topology is ( {0,1} , the empty set ). Limit pt. compact means every infinite subset has a limit point, and countably compact means every countable open cover contains a finite subcover.

If somebody can help with hints, I'd be much obliged. I've tried drawing pictures, and some of the other parts of the questions I've already gotten, but this has been a hard class in general for me because of the sheer volume of definitions and theorems. The professor helps very much when I go to his office, some problems he practically does for us, but I'm hardly ever free when he has office hours. I usually work all day until 9 or 10:00 Thursday and Friday night to get it turned in on Friday (he has a "slide it under my door" policy). I can't work Monday or Tuesday on it because I have a job that takes alot of my time, plus I have to work on my other classes SOMETIME, and that sometime is monday and tuesday. I'm feeling a little more behind than usual this week, and I'll have to share his one hour of office time with alot of other students tomorrow. Please post any ideas you may have, any hint is better than none.

Thanks in advance for any help. All hints are greatly appreciated.
Aaron

Last fiddled with by synergy on 2004-11-05 at 02:01 Reason: needed a small change
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Old 2004-11-05, 06:02   #2
Zeta-Flux
 
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May 2003

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synergy,

Sounds like your homework. ;)

Quote:
I also need to show whether [0,1]^|R, that is, [0,1]x[0,1]x... an uncountable number of times, whether that is sequentially compact, i.e. whether EVERY sequence has a convergent subsequence.
I'm assuming that your are using the product topology. In that case, the product of compact spaces is compact by Tychonoff's theorem, and in particular is sequentially compact.

Quote:
I also need to show that X x Y is limit point compact but not countably compact, where X=the natural numbers, the topology on X is the power set, Y={0,1} and its topology is ( {0,1} , the empty set ).
Let Z = X x Y. Let L = {z_1, z_2,...} be an infinite sequence of points in Z. We can write z_i=(x_i, y_i) for x_i in X and y_i in Y. Without loss of generality, we may assume that y_1=0. Let z=(x_1, 1). Then ANY closed set containing z will also contain z_1, and hence z is a limit point of L. (Basically, you need to use the fact that Y is not a T_1 space here.)

I'll let you ponder the countable compactness questions a bit more.

Cheers,
Zeta-Flux
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Old 2004-11-05, 17:18   #3
synergy
 
Aug 2004

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Thanks a bunch, Zeta-Flux.
Yeah, it's homework. I never seem to have enough time, that's the big problem. This'll help alot.
Thanks again.
Aaron
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Old 2004-11-05, 17:21   #4
synergy
 
Aug 2004

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Stupid me , Tychonoff's theorem was in one of the hints the professor posted!
Aaron
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