Is the interior of a compact subet of C^0 empty?

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February 24th 2010, 10:11 AM
southprkfan1

Is the interior of a compact subet of C^0 empty?

This question is very similar to one I asked here: http://www.mathhelpforum.com/math-he...ons-empty.html

Let C0 be the set of continuous functions (from [a,b] to R) and a subspace of Cb, the set of bounded functions (from [a,b] to R), defined with the supnorm metric.

Let A be a compact subset of C0.

Is the Interior of A empty?

My first thought was that the proof is the same as in the link above (that is, yes it is empty since we can easily find a function g in any open ball around f in A where g is not continuous), but I'm worried this doesn't apply here because the subset A may only contain functions with certain properties, and so we don't know if g is in A.

Am I making any sense here? I think I successfully confused myself.
February 24th 2010, 12:46 PM
Opalg

Quote:

Originally Posted by southprkfan1

This question is very similar to one I asked here: http://www.mathhelpforum.com/math-he...ons-empty.html

Let C0 be the set of continuous functions (from [a,b] to R) and a subspace of Cb, the set of bounded functions (from [a,b] to R), defined with the supnorm metric.

Let A be a compact subset of C0.

Is the Interior of A empty?

My first thought was that the proof is the same as in the link above (that is, yes it is empty since we can easily find a function g in any open ball around f in A where g is not continuous), but I'm worried this doesn't apply here because the subset A may only contain functions with certain properties, and so we don't know if g is in A.

Am I making any sense here? I think I successfully confused myself.

As stated here, the answer is exactly the same as in the previous question: you can easily find a discontinuous function in any neighbourhood of a given continuous function.

However, this problem, unlike the previous one, makes sense (and is much more interesting) if you work entirely within C[a,b], the space of continuous functions on [a,b]. The reason is that this time the set A is not the whole space but a compact subset of it. So it makes sense to ask if A can contain an open subset of C[a,b]. I think that the answer is no, and that the reason is the Arzelà–Ascoli theorem, which characterises the compact subsets of C[a,b]. I don't have time to think it through carefully just now, but I think that it is not possible for an open subset of C[a,b] to be equicontinuous. It follows from the A–A theorem that such a set can never be compact in C[a,b].
February 24th 2010, 01:54 PM
Drexel28

Quote:

Originally Posted by Opalg

As stated here, the answer is exactly the same as in the previous question: you can easily find a discontinuous function in any neighbourhood of a given continuous function.

However, this problem, unlike the previous one, makes sense (and is much more interesting) if you work entirely within C[a,b], the space of continuous functions on [a,b]. The reason is that this time the set A is not the whole space but a compact subset of it. So it makes sense to ask if A can contain an open subset of C[a,b]. I think that the answer is no, and that the reason is the Arzelà–Ascoli theorem, which characterises the compact subsets of C[a,b]. I don't have time to think it through carefully just now, but I think that it is not possible for an open subset of C[a,b] to be equicontinuous. It follows from the A–A theorem that such a set can never be compact in C[a,b].

I agree with Opalg and will work on it as well. I think I've done a problem before about open subsets of $\mathcal{C}\left[X,\mathbb{R}$ for compact metric spaces $X$ . Let me look for it.