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

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In summary, there is a problem with a limit point compact subspace X in a Hausdorff space Z, as Munkres asks if it is closed. A possible counterexample can be found in the space of linearly ordered ordinals. It is necessary to exclude the case of a metric space or a compact Hausdorff space, as limit point compactness and compactness are equivalent in these cases. It is possible that the counterexample exists even if Z is compact, as a compact set X in a Hausdorff space Z is not necessarily closed.
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facenian
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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?
 
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Are you familiar with the space of linearly ordered ordinals? It provides a counterexample to your claim.
 
  • #3
Math_QED said:
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?
 
  • #4
facenian said:
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
 
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facenian said:
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.
 
  • #6
WWGD said:
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.
 
  • #7
facenian said:
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.
 
  • #8
WWGD said:
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.
 
  • #9
facenian said:
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.
 

1. What is a limit point compact subspace in a Hausdorff space?

A limit point compact subspace in a Hausdorff space is a subset of a Hausdorff space that is both limit point compact and Hausdorff. This means that every sequence in the subspace has a limit point, and the subspace satisfies the Hausdorff property, which states that for any two distinct points in the subspace, there exist disjoint open sets containing each point.

2. Why is the concept of limit point compactness important in topology?

The concept of limit point compactness is important in topology because it allows us to study the behavior of sequences and the structure of a space. It also has applications in areas such as analysis and geometry, and is a useful tool in proving theorems and solving problems in these fields.

3. How is limit point compactness related to other types of compactness?

Limit point compactness is a weaker form of compactness compared to other types, such as sequential compactness or compactness in the sense of open covers. This means that not all limit point compact spaces are necessarily sequentially compact or compact in the sense of open covers, but the converse is true. In other words, every sequentially compact or compact in the sense of open covers space is also limit point compact.

4. Can a limit point compact subspace be non-Hausdorff?

No, a limit point compact subspace must also satisfy the Hausdorff property. This is because the definition of limit point compactness includes the requirement that every sequence has a limit point, and this can only be guaranteed in a Hausdorff space. Therefore, a non-Hausdorff subspace cannot be limit point compact.

5. How is limit point compactness different from local compactness?

Limit point compactness and local compactness are two different concepts in topology. While limit point compactness is a global property that applies to the entire space, local compactness is a local property that applies to each point in the space. A space is locally compact if every point has a compact neighborhood, while a space is limit point compact if every sequence has a limit point. In general, a locally compact space may not be limit point compact, and vice versa.

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