How to prove something is closed and bounded, ie compact

In summary, the task is to prove that a closed ball with a radius of r around point x0 is both closed and bounded. The definitions of "closed ball," "sphere," "closed," and "bounded" must be given in order to begin the proof. The book being used does not provide helpful information and online searches have not yielded much. It is important to determine which definition of "closed" the book is using, as there are multiple possibilities. Additionally, proving something is bounded means that it can fit inside a ball with a finite radius, and this is not related to the concept of a boundary.
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Homework Statement


I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0).


Homework Equations





The Attempt at a Solution


How does one go about proving something is closed and bounded? My book is not very helpful and searching hasn't yielded much. This is only a part of the problem, but the rest should be doable once I get this.
 
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Start by giving the definitions of "closed ball", "sphere", "closed" and "bounded".
 
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Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started
 
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MeMoses said:
Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started

What book are you using? Aren't those things in there?
 
  • #5
Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?
 
  • #6
MeMoses said:
Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?
That's not what bounded means. In fact, bounded has nothing to do with boundary. A bounded set is one that is fits inside a ball of some finite radius.

There are various equivalent definitions of closed. You must find out which one your book is using. Some possibilities:
* a set is closed if and only if its complement is open
* a set is closed if and only if it contains all of its limit points
* a set is closed if and only if it contains all of its boundary points
 

1. How do you define a closed set?

A closed set is a set that contains all of its limit points. In other words, every sequence of points within the set must have a limit point that is also within the set.

2. What is the definition of a bounded set?

A bounded set is a set that has a finite or defined limit on its size or extent. In other words, there is a finite distance between any two points within the set.

3. How do you prove that a set is closed and bounded?

To prove that a set is closed and bounded, you must show that it satisfies both the definitions of closed and bounded. This can be done by showing that all limit points are contained within the set, and that there is a finite distance between any two points within the set.

4. What is the significance of a set being compact?

A set being compact means that it is both closed and bounded. This has important implications in mathematics and physics, as compact sets have nice properties that make them easier to work with and analyze.

5. Can a set be compact in one space but not in another?

Yes, a set can be compact in one space but not in another. This is because the definition of compactness depends on the specific space in which the set is being analyzed. A set may satisfy the definition of compactness in one space, but not in another.

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