How to prove something is closed and bounded, ie compact

  • Thread starter MeMoses
  • Start date
  • #1
MeMoses
129
0

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.
 

Answers and Replies

  • #2
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,305
Start by giving the definitions of "closed ball", "sphere", "closed" and "bounded".
 
  • #3
MeMoses
129
0
Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started
 
  • #4
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,178
3,305
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
MeMoses
129
0
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
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,475
257
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
 

Suggested for: How to prove something is closed and bounded, ie compact

Replies
12
Views
411
Replies
4
Views
421
  • Last Post
Replies
18
Views
438
Replies
12
Views
456
Replies
2
Views
197
Replies
7
Views
342
Replies
0
Views
237
Replies
7
Views
1K
Replies
2
Views
779
Top