I Help Solve Munkres Question: Limit Point Compact Subspace in Hausdorff Space

[facenian]

TL;DR Summary

Problem with a limit point compact space

Hello, I have a problem with a question in Munkres topology book. Munkres asks if a limit point compact subspace X in a Hausdorff space Z is closed.
I tried to prove it by contradiction by taking an infinite set in X and suppose it has a limit point that is not in X, however, I could not find a contradiction. I suspect that an example exists where X is not closed. Can somebody please help?

 

[member 587159]

Are you familiar with the space of linearly ordered ordinals? It provides a counterexample to your claim.

 

[facenian]

Are you familiar with the space of linearly ordered ordinals? It provides a counterexample to your claim.

Sorry, I don't understand your answer. My claim is that I suspect that there exists a Hausdorff space in which a limit point subspace is not close. Do you mean that I can find a counterexample in the linearly ordered ordinals that confirms my suspicion?

 

[member 587159]

Sorry, I don't understand your answer. My claim is that I suspect that there exists a Hausdorff space in which a limit point subspace is not close. Do you mean that I can find a counterexample in the linearly ordered ordinals that confirms my suspicion?

Exactly. See e.g. here:

https://math.stackexchange.com/q/1404003/661543

 

[WWGD]

Summary: Problem with a limit point compact space

Hello, I have a problem with a question in Munkres topology book. Munkres asks if a limit point compact subspace X in a Hausdorff space Z is closed.
I tried to prove it by contradiction by taking an infinite set in X and suppose it has a limit point that is not in X, however, I could not find a contradiction. I suspect that an example exists where X is not closed. Can somebody please help?

Just to narrow it down, hopefully get closer to an answer, you need to exclude metric, and compact Hausdorff ( of course, metric spaces are Hausdorff, so there is overlap). So it comes down to, if possible, to a setting in which limit-point compactness does not imply compactness. Hope I did not just tell you something you knew.

 

[facenian]

Just to narrow it down, hopefully get closer to an answer, you need to exclude metric, and compact Hausdorff ( of course, metric spaces are Hausdorff, so there is overlap). So it comes down to, if possible, to a setting in which limit-point compactness does not imply compactness. Hope I did not just tell you something you knew.

Yes, in a metric space limit point compactness is equivalent to compactness, so a counterexample cannot be found there. However, I don't know why you also want the exclude the case when Z is compact.

 

[WWGD]

Yes, in a metric space limit point compactness is equivalent to compactness, so a counterexample cannot be found there. However, I don't know why you also want the exclude the case when Z is compact.

Because compact+Hausdorff $\rightarrow$ closed.

 

[facenian]

Because compact+Hausdorff $\rightarrow$ closed.

A compact set X in a Hausdorff space Z is closed however, there could exist(I believe) a limit point compact set that it is not compact which is not closed and it is inside a Z compact Hausdorff space, i.e., the counterexample could exist whether Z is compact or not.

 

[WWGD]

A compact set X in a Hausdorff space Z is closed however, there could exist(I believe) a limit point compact set that it is not compact which is not closed and it is inside a Z compact Hausdorff space, i.e., the counterexample could exist whether Z is compact or not.

Yes, that is what I was trying to get at, we need to find a setting where the two do not coincide.