Prove that the boundary of S is compact

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December 19th 2012, 03:09 AM
Plato13

Prove that the boundary of S is compact

Let S be a compact subset of a hausdorff space X. Prove that the boundary of S is compact.
December 19th 2012, 03:48 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

Let S be a compact subset of a hausdorff space X. Prove that the boundary of S is compact.

Have you proved that in a Hausdorff space boundaries of sets are closed?

What is known about closed subsets of compact sets?
December 19th 2012, 03:59 AM
Plato13

Re: Prove that the boundary of S is compact

No I haven't proved that. That proof would be helpful.
December 19th 2012, 04:15 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

No I haven't proved that. That proof would be helpful.

Well then prove that the boundary is a closed set. Show it contains all its limit points.
December 19th 2012, 04:59 AM
Plato13

Re: Prove that the boundary of S is compact

Let S be a subset of a hausdorff space X and let B be it's boundary. Let x be a limit point of B. Then every neighbourhood of x contains an element y in B different to x. So there exists disjoint open sets U,V such that x is in U, y is in V.
Now i'm stuck.
December 19th 2012, 06:27 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

Let S be a subset of a hausdorff space X and let B be it's boundary. Let x be a limit point of B. Then every neighborhood of x contains an element y in B different to x.

Let's use $\beta(S)$ for the boundary of $S$ .

What is the exact meaning of $t\in\beta(S)$ . (i.e. what is a boundary point?)

Note how I edited your reply. Say $U$ is a neighborhood of x.
Then there is a neighborhood of y, $V$ , such that $x\not\in V\subset U.$ .

Now apply the definition of a boundary point.
December 19th 2012, 07:08 AM
Plato13

Re: Prove that the boundary of S is compact

Why do you say V is a subset of U? Also I don't know much about x being a boundary point except the boundary is closure/interior.
December 19th 2012, 07:28 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

Why do you say V is a subset of U? Also I don't know much about x being a boundary point except the boundary is closure/interior.

That is your whole problem in a nutshell.
Let this be a lesson to you. Always post what you know about the question. I had no way of knowing that you were working with what I consider a non-standard definition.

Notation: $\mathcal{I}(S)$ stands for the interior of $S$ .

By your definition $\beta(S)=\overline{S}\setminus\mathcal{I}(S)$ .

But $\mathcal{I}(S)$ is open set. Therefore its complement is closed.

Thus $\beta(S)$ is the intersection of two closed sets. SO?
December 19th 2012, 07:33 AM
Plato13

Re: Prove that the boundary of S is compact

I'm sorry about that, I will bear your advice in mind in future. Where have you used X is hausdorff?
December 19th 2012, 07:49 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

Where have you used X is hausdorff?

It was not use. Being Hausdorff is not necessary.

BTW: It is Hausdorff. That is a proper name.
December 19th 2012, 08:01 AM
Plato13

Re: Prove that the boundary of S is compact

Are you sure you've proved the boundary is compact, and not closed? Something's wrong because it is a necessary hypothesis

BTW: are you highlighting the capital or that it should be droff and not dorff?
December 19th 2012, 08:24 AM
Plato

Re: Prove that the boundary of S is compact

Quote:

Originally Posted by Plato13

Are you sure you've proved the boundary is compact, and not closed? Something's wrong because it is a necessary hypothesis.

What was shown is that $\beta(S)$ is a closed set.
Any closed subset of a compact set in a Hausdorff space is compact.
Thus $\beta(S)\subseteq\overline{S}$ is compact.
December 19th 2012, 08:38 AM
Plato13

Re: Prove that the boundary of S is compact

Thanks for your help. In future please don't respond to my questions as other (better) helpers will see you're dealing with it and I will be stuck with your minimalist answers. I know your ethos is let the student do it themselves but sometimes it is more instructive for me to see the proof.