Compact Space Hausdorff Preservation

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Discussion Overview

The discussion revolves around the question of whether it is possible to transform a compact space into a Hausdorff space while preserving its compactness. Participants explore various approaches, examples, and theoretical implications related to this topic.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant asks if there is a process to make a compact space Hausdorff while preserving compactness, seeking a method akin to "hausdorffication."
  • Another participant points out that a sphere is an example of a compact and Hausdorff space, but this does not address the original question.
  • A participant suggests that if the boundary of a compact set is a manifold, a continuous mapping might extend the set into itself, though they express doubt about applying this to all compact sets, particularly finite sets.
  • One participant questions what aspects need to be preserved and mentions identifying inseparable points as a potential method, noting its effectiveness in some cases but not in others, such as the Zariski plane.
  • A later reply indicates that enlarging the topology of a compact space to make it Hausdorff is generally not possible.
  • Another participant humorously suggests that removing all points but one from any topological space results in a Hausdorff and compact space, though this may not be a practical solution.
  • Clarification is provided that the original inquiry pertains to embedding a compact space in a way that results in a Hausdorff space.
  • One participant asserts that altering the topology to achieve both properties is not feasible, questioning if a Hausdorff space can contain a compact non-Hausdorff subspace, to which they respond negatively.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of transforming a compact space into a Hausdorff space while preserving compactness. There is no consensus on a definitive method or solution, and multiple competing perspectives are presented.

Contextual Notes

Participants acknowledge limitations in their proposals, particularly regarding the applicability of certain methods to all compact spaces and the implications of altering topological properties.

fallgesetz
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Is there a way to make a compact space hausdorff while preserving compactness?
 
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Specify what you mean.
A sphere is compact and Hausdorff.
 
I am not looking for an example of a compact & hausdorff space.

I am asking -- if I have a compact space, is there a process by which I can make it a hausdorff space while preserving compactness.

In a sense, I am looking for something like one point compactification(but not that, more like "hausdorffication").
 
Just a thought... if the border of your compact set is a manifold, I think you can usually find a continuous mapping that extends the set into itself when going over the border. But since for example finite sets are compact, I don't see a sane way of making every compact set Hausdorff.
 
What are you looking to preserve? What is the application?

There are certainly things you can do: e.g. you can identify inseparable points. This works well for something like a Euclidean line with a double point at the origin (which gives you the Euclidean line). This doesn't work well for something like the Zariski plane over a field. (the result is the one-point space).
 
0xDEADBEEF said:
But since for example finite sets are compact...

That statement sounds stupid in retrospective sorry...
 
Perhaps he means this: Given a compact space X (not Hauseorff), can you enlarge the topology to make it a compact Hausdorff space? If that is what he means, then the answer is, in general, "no".
 
Here is a way:

Take any topological space. Remove all points but one. This space is Hausdorff and compact.
 
Ok, good point.

I should clarify that originally I was asking for some embedding of a compact space which turns out to be hausdorff.
 
  • #10
This is a topological property. You can't alter either of those without altering the topology. Do you mean: is there a Hausdorff space which admits a compact non-Hausdorff subspace? The answer to that is no.
 

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