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