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Question about Frechet compactness

  • Thread starter facenian
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  • #1
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Homework Statement


Let ##X## be a limit point compact space. If ##X## is a subspace of the Housdorff space ##Z##, does it follow that ##X## is closed in Z?

Homework Equations


A space ##X## is said to be limit point compact if every infinite subset of ##X## has a limit point.

The Attempt at a Solution


If ##X## is finite it is closed so suppose it is infinite then it has a limit point in ##X## however this does not exclude the posibility of ##X## having a limit point in ##Z## outside ##X## which implies that it is not closed in ##Z##.
Is this correct? In which case a counter example is needed.
 

Answers and Replies

  • #2
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The statement is false. Take ##Z = \mathbb{R}## with the usual topology. Then ##Z## is Hausdorff as it is metrizable and ##X = (0,1)## is not closed in ##Z##, while it is limit point compact as subspace (by the Bolzano-Weierstrass theorem)
The problem is that this is not a counter example. The infinte set ##\{1/(n+1):n\in Z^+\}\subset (0,1)## does not have a limit point in (0,1) so this space is not limit point compact. Given the Heine-Borel theorem for ##R^n## a counter example can exists only outside ##R^n## with the usual topology or the fact that limit point compactness and compactnes are equivalent in a metric space.
 
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  • #3
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The problem is that this is not a counter example. The infinte set ##\{1/(n+1):n\in Z^+\}\subset (0,1)## does not have a limit point in (0,1) so this space is not limit point compact. Given the Heine-Borel theorem for ##R^n## a counter example can exists only outside ##R^n## with the usual topology or the fact that limit point compactness and compactnes are equivalent in a metric space.
You are right. This has been a long day for me haha. Currently studying for a functional analysis exam. I deleted my post so that it seems that you didn't get any responses (I will delete this post too)
 
  • #4
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You are right. This has been a long day for me haha. Currently studying for a functional analysis exam. I deleted my post so that it seems that you didn't get any responses (I will delete this post too)
I believe that wrong answers are also instructive though and it is not good to delete them since they may help elucidating a correct answer.
 
  • #5
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