Confusion with Disconnected sets

  • Context: Graduate 
  • Thread starter Thread starter Mr-T
  • Start date Start date
  • Tags Tags
    Confusion Sets
Click For Summary

Discussion Overview

The discussion revolves around the concept of connectedness in the context of metric topology, specifically examining the set X = (0,1] ∪ (1,2) and whether it can be considered connected under the usual metric topology of the reals.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about why the set X = (0,1] ∪ (1,2) is considered connected, referencing a specific definition of disconnectedness involving open sets.
  • Another participant points out that the open sets G and V must be open, implying that the initial definition may need clarification.
  • A further reply suggests that the sets (0,1] and (1,2) cannot serve as the required open sets since (0,1] is not open, and notes that (0,1] ∪ (1,2) simplifies to (0,2).

Areas of Agreement / Disagreement

Participants appear to agree on the need for open sets in the definition of disconnectedness, but there is disagreement regarding the connectedness of the specific set X and the appropriate open sets to use in the analysis.

Contextual Notes

The discussion highlights the importance of the definitions used in topology, particularly the distinction between open and closed sets, and the implications for connectedness. There may be missing assumptions regarding the properties of the sets involved.

Mr-T
Messages
21
Reaction score
0
Hello,

I am having some difficulties understanding why a subset under the usual metric topology of the reals is connected.

How can a set X = (0,1] u (1,2) be connected?

The definition I am using is:

A is disconnected if there exists two open sets G and V and the following three properties hold:

(1) A intersect G ≠ ∅
A intersect V ≠ ∅

(2) A is a proper subset of the union of G and V.

(3) the intersection of G and V is the empty set.

Thanks
 
Last edited:
Physics news on Phys.org
And you also want ##G## and ##V## to be open. No?
 
Yea, that would be more correct.
 
Okay, so can you find two such open sets for (0, 1]\cup (1, 2)? (0, 1] and (1, 2) will not do because (0, 1] is not open. (And, did you notice that (0, 1]\cup (1, 2)= (0, 2)?)
 

Similar threads

High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Confused by proof in Hocking and Young
  • · Replies 4 ·
Replies
4
Views
3K
MHB The Set of Borel Sets .... Axler Pages 28-29 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad The Set of Borel Sets .... Axler Pages 28-29 .... ....
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad An open set can be a closed interval?
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Homemorphism in quotient topology
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Is [0,1] under the subspace topology Hausdorff, Compact, or Connected?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Trying to understand why a set is both open and closed
  • · Replies 7 ·
Replies
7
Views
2K
MHB Disconnected Sets .... Palka, Lemma 3.1 ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Topology Words: Reasons for the particular names
  • · Replies 2 ·
Replies
2
Views
2K