Compact Space Hausdorff Preservation

[fallgesetz]

Is there a way to make a compact space hausdorff while preserving compactness?

 

[0xDEADBEEF]

Specify what you mean.
A sphere is compact and Hausdorff.

 

[fallgesetz]

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").

 

[0xDEADBEEF]

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.

 

[Hurkyl]

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]

But since for example finite sets are compact...

That statement sounds stupid in retrospective sorry...

 

[g_edgar]

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".

 

[Jamma]

Here is a way:

Take any topological space. Remove all points but one. This space is Hausdorff and compact.

 

[fallgesetz]

Ok, good point.

I should clarify that originally I was asking for some embedding of a compact space which turns out to be hausdorff.

 

[zhentil]

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.