Any compact subset is a contained in finite set + a convex set?

In summary, the homework statement is that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0.
  • #1
Fractal20
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1

Homework Statement


So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything to support this. Where does this come from?

Homework Equations





The Attempt at a Solution


I keep thinking that maybe it can be approached by the set being totally bounded since it is compact. So it seems for a given ε can cover the subset in open balls of radius ε. Then the centers of these balls is a finite set, then somehow choosing a corresponding convex set might let you represent any of the elements of the original compact set as the sum of elements of these new sets. But I don't know, I just don't have any basis from which to approach this.
 
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  • #2
I would expect that you can prove it like this (just more formal):
Assume there is a compact set which cannot be described as a subset of a convex set and a finite set. This would require an infinite number of elements "far away" (without bound). You can cover each of this element with an open set (using an infinite amount of them), such that the only covering subset is the whole set of those open sets. Therefore, there is no finite subcover.
 
  • #3
So if they weren't unbounded, then you could cover it with a ball of radius bigger than the largest element? I think I am okay with this, but in the proof where this was stated, it was for an convex neighborhood about 0 which is a subset of any arbitrary open neighborhood of 0. So in turn it seems like the convex neighborhood is really arbitrary. So then it seems like the convex set wouldn't be free to be large necessarily. For example, say it is a closed ball in a finite dimensional space, it still seems counter-intuitive that the closed ball is the subset of a finite set and an arbitrary convex neighborhood. Thanks for the help!
 
  • #4
So if they weren't unbounded, then you could cover it with a ball of radius bigger than the largest element?
Sure

I have no idea what you mean in the rest of the post.
 
  • #5
Haha, sorry. So this all comes up in a proof for "the closed convex hull of a compact subset of a completely metrizable space is compact". It can be found on page 186 here:

http://books.google.com/books?id=4h...epage&q=closed convex hull is compact&f=false

Anyway, the very first part is that for any arbitrary open neighborhood U about 0 there exists a convex neighborhood of 0, V which is a subset U and such that the compact subset in question is a subset of a finite set plus this convex set. But at the end of the proof they shrink U since it was arbitrary, so then it seems even more surprising. I don't know if that made sense at all?
 

1. What is a compact subset?

A compact subset is a subset of a topological space that is both closed (contains all of its limit points) and bounded (can be contained within a finite distance). In other words, it is a set that is not missing any of its points and is not infinitely spread out.

2. What does it mean to be contained in a finite set?

A set is considered to be contained in a finite set if all of its elements are also elements of the finite set. This means that the set has a limited number of elements and is not infinite.

3. What is a convex set?

A convex set is a set where, for any two points in the set, the line connecting them lies completely within the set. In other words, the set does not have any holes or concave areas.

4. Why is it important for a compact subset to be contained in a finite set and a convex set?

It is important for a compact subset to be contained in a finite set and a convex set because these properties ensure that the subset is well-behaved and can be easily studied and analyzed. The finite set property guarantees that the subset is not too spread out, while the convex set property guarantees that the subset does not have any irregularities or holes.

5. How does the concept of compactness relate to real-world applications?

The concept of compactness is widely used in various fields of science and engineering, such as optimization, control theory, and topology. It allows for the study and analysis of sets that have a finite number of elements and are well-behaved, making it a useful tool in solving real-world problems.

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