Closure, Compactness, and Completeness

  • Thread starter Thread starter LauraLovies
  • Start date Start date
  • Tags Tags
    closure
Click For Summary

Homework Help Overview

The discussion revolves around the properties of the set S = [0,1] U {3}, specifically whether it is closed, open, compact, complete, or connected.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definitions of open and closed sets, with one participant expressing uncertainty about the classification of the set. There is an attempt to analyze the connectivity of the set and its implications for compactness and completeness.

Discussion Status

Some participants are exploring the definitions of open and closed sets, while others are questioning the connectivity of the set. Guidance has been offered to start with definitions, indicating a productive direction in the discussion.

Contextual Notes

Participants are navigating the definitions and properties of sets in the context of topology, with an emphasis on understanding the implications of being open, closed, compact, complete, or connected.

LauraLovies
Messages
12
Reaction score
0

Homework Statement



The set S = [0,1] U {3}

Homework Equations




I need to say whether it is closed, open, compact, complete or connected. If it is not compact, give an example why. Same thing for completeness. If its not connected, state why not.

The Attempt at a Solution


I think it is definitely not connected. I also think the set is closed and therefore bounded and compact and complete. But I could be way off...
 
Physics news on Phys.org
as always start from the definitions...
 
yes i know...but am i on the right track. i just can't seem to pinpoint whether the set is open or closed. and if i knew that...then i would be fine...
 
ok so start with the definition of open and/or closed...

clearly any neighbourhood of 3 conatins points not in theset, so it cannot be open
 

Similar threads

Prove that the intersection of subspaces is compact and closed
  • · Replies 12 ·
Replies
12
Views
3K
A proof that a union of compact spaces is compact
  • · Replies 12 ·
Replies
12
Views
3K
Are the following Sets: Open, Closed, Compact, Connected
  • · Replies 3 ·
Replies
3
Views
2K
Is Every Connected Metric Space Compact?
  • · Replies 1 ·
Replies
1
Views
2K
Path-Connected Sets and Their Closures: A Counterexample?
  • · Replies 1 ·
Replies
1
Views
2K
Exploring Open, Closed, Bounded, and Compact Sets in R with a Unique Metric
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Compact manifold cannot be represented by (single) parametric equation
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Tough lemma on locally finite refinement
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Another similar notion to compactness
  • · Replies 2 ·
Replies
2
Views
559
Undergrad On two definitions of locally compact
  • · Replies 5 ·
Replies
5
Views
3K