Discussion Overview
The discussion revolves around a question from Munkres' topology book regarding whether a limit point compact subspace X in a Hausdorff space Z is necessarily closed. Participants explore the implications of limit point compactness and seek counterexamples to support or refute the claim.
Discussion Character
Main Points Raised
- One participant expresses difficulty in proving that a limit point compact subspace X in a Hausdorff space Z is closed, suspecting that a counterexample exists where X is not closed.
- Another participant suggests that the space of linearly ordered ordinals could serve as a counterexample to the claim that limit point compactness implies closedness.
- There is a reiteration of the need to exclude metric and compact Hausdorff spaces from consideration, as limit point compactness is equivalent to compactness in metric spaces.
- Participants discuss the implications of compactness in Hausdorff spaces, noting that a compact set is closed, but questioning whether a limit point compact set that is not compact could still be closed within a compact Hausdorff space.
- There is a consensus that the discussion needs to focus on settings where limit point compactness does not imply compactness, but no specific examples are agreed upon.
Areas of Agreement / Disagreement
Participants generally agree that limit point compactness does not imply closedness in certain contexts, but there is no consensus on specific examples or the necessity of excluding certain types of spaces.
Contextual Notes
Limitations include the dependence on definitions of compactness and Hausdorff spaces, as well as the unresolved nature of the examples discussed.