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## Homework Statement

So I had a homework problem which was to show that a certain function space was compact via Ascoli-Arzela Theorem. I was okay with doing this, accept, it appears the corresponding uniformly convergent sequence I found in any infinite set need not converge to something in the set. I brought this up with my professor and was told that outside of finite Euclidean space a compact set does not need to be closed.

So I tried to dig up different definitions of compact because I this seemed to be the opposite of what I remember and everything seemed to say that the convergent subsequence needs to converge to something in the set for it to be compact.

So my real question is, is this true? A compact function space need not be closed? If it helps the original question is:

Let M be a bounded subset of C[a,b] show the set {R(x) =∫

_{a}

^{x}f(t) dt | f is in M} is compact.

So I am fine with showing that this fits the criteria for Ascoli-Azela but again it seems like I can construct convergent sequences that don't converge to something in the set. For example, suppose M = {fn| fn = 1/n for n=1,2,3...}. Then the corresponding set {R} is equal to the series I will call Rn = (1/n)(x-a). This converges uniformly to the function f = 0, but this is not in {R}.

Again, my question is if compact spaces need not be closed. If so, when is this true? Thanks!