Need help with 'Compact set Prove' question

  • Thread starter Thread starter pantin
  • Start date Start date
  • Tags Tags
    Set
Click For Summary

Homework Help Overview

The discussion revolves around the compactness of the union of line segments in R², specifically the set S formed by the segments L_i between points (-1/i, 1/i) and (1/i, 1/i) for i=1,2,3.... Participants are tasked with justifying whether this union is compact.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the properties of the set S, questioning whether the line segments include their endpoints and discussing the implications of this on the compactness of the union. There is a focus on whether the set is closed and bounded, as well as the identification of limit points.

Discussion Status

The discussion is active, with participants examining the definitions of closed and bounded sets in relation to compactness. Some have provided insights into the nature of the endpoints and limit points, while others are still seeking clarification on how to proceed with the proof.

Contextual Notes

There is uncertainty regarding the inclusion of endpoints in the line segments, which affects the determination of whether the union S is closed. The participants are also considering the implications of limit points in their analysis.

pantin
Messages
20
Reaction score
0
Let L_i denote the line segment in R^2 between the points (-1/i, 1/i) and (1/i, 1/i) with i=1,2,3...
Is the union S=U_(i=1 to infinite)L_i compact? Justify.




after I drew the graph, I just noticed that this set is not closed, because all the start points and end points of those segment are not included, so why need to prove?
 
Physics news on Phys.org
pantin said:
Let L_i denote the line segment in R^2 between the points (-1/i, 1/i) and (1/i, 1/i) with i=1,2,3...
Is the union S=U_(i=1 to infinite)L_i compact? Justify.




after I drew the graph, I just noticed that this set is not closed, because all the start points and end points of those segment are not included, so why need to prove?

None of those open intervals contains its endpoints certainly in that segment. You need to show that the UNION of all those intervals is an interval that does not include its endpoints. That is, I think, true and easy to prove.
 
Where does it say the line segments don't include the endpoints? Are you sure they don't? If they don't, then S is clearly not closed, since the endpoints aren't in any of the L_i. But what if the line segments do include the endpoints?
 
oh yes! haha, you are smart, it doesn't say the line segments do not include endpoints,that's just the coordinate sign!

but then how to continue..
 
A subset of R^2 is compact only if it's closed and bounded. S is clearly bounded, so we just need to check if it's closed. This means it contains all it's limit points.
You've drawn the graph. Does it look like there's a limit point that isn't in any of the intervals L_i?
 
i got it, it doesn't contain 0 , which is the limit of the sequence

thx a lot
 

Similar threads

Prove that the intersection of subspaces is compact and closed
  • · Replies 12 ·
Replies
12
Views
3K
Exploring Open, Closed, Bounded, and Compact Sets in R with a Unique Metric
  • · Replies 1 ·
Replies
1
Views
2K
Distance of a point from a compact set in ##\Bbb{R}##
  • · Replies 14 ·
Replies
14
Views
3K
Prove that Locally Lipschitz on a Compact Set implies Lipschitz
  • · Replies 4 ·
Replies
4
Views
8K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Tough lemma on locally finite refinement
  • · Replies 8 ·
Replies
8
Views
3K
Prove a subset of R^n is compact
  • · Replies 4 ·
Replies
4
Views
2K
Proving intersection of finitely many open sets is open
  • · Replies 1 ·
Replies
1
Views
2K
Infinite union of closed sets is not closed
  • · Replies 13 ·
Replies
13
Views
5K
Proving Compact Set Exists with m(E)=c
  • · Replies 11 ·
Replies
11
Views
3K