Is \{1,2,3,4,5\ldots\} a Closed Set in \mathbb{R}?

  • Context: Undergrad 
  • Thread starter Thread starter AxiomOfChoice
  • Start date Start date
  • Tags Tags
    Closed Set
Click For Summary

Discussion Overview

The discussion revolves around whether the set \{1,2,3,4,5\ldots\} (the set of positive integers) is a closed set in the context of real numbers (\mathbb{R}). The scope includes definitions of closed and open sets, boundary considerations, and the properties of complements in topology.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the set \{1,2,3,4,5\ldots\} might be closed but expresses uncertainty.
  • Another participant states that the definition of a closed set involves the complement being open and questions whether the complement of the set is open.
  • A request is made for a demonstration that the complement of the set is open in \mathbb{R}.
  • One participant explains that a closed set contains its boundary and argues that the set is closed, asserting that it is its own boundary.
  • A reiteration of the definition of closed sets is made, confirming that the union of open sets is indeed open.

Areas of Agreement / Disagreement

Participants express differing views on whether the set is closed, with some supporting the idea that it is closed based on boundary definitions, while others focus on the complement's properties. The discussion remains unresolved.

Contextual Notes

There are limitations in the discussion regarding the definitions of closed and open sets, as well as the handling of boundaries and complements, which are not fully explored or resolved.

AxiomOfChoice
Messages
531
Reaction score
1
The set \{1,2,3,4,5\ldots\}...is it closed as a subset of \mathbb{R}? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.
 
Physics news on Phys.org
The definition of closed is the complement of an open set. So is the set

( - \infty,1) \cup (1,2) \cup (2,3) \cup (3,4)... open?
 
Can you show that its complement is open in \mathbb{R}?
 
A closed set contains its bondary.
The boundary are al the points where you can draw an arbitrarily small circle around it and there are always points inside and outside the set. So yes your set is closed and it is its own boundary.
 
Last edited:
Office_Shredder said:
The definition of closed is the complement of an open set. So is the set

( - \infty,1) \cup (1,2) \cup (2,3) \cup (3,4)... open?
Yes. Any union of open sets is open. Thanks! :)
 

Similar threads

High School Sequential Compactness of Sets: A Definition
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Partial derivative of Dirac delta of a composite argument
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Questions on ##\mathbb{R}##
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How do you define unboundedness in Euclidean space?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Relating integration of forms to Riemann integration
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad About the definition of topological manifold using closed sets
  • · Replies 14 ·
Replies
14
Views
10K
Undergrad Gradient With Respect to a Set of Coordinates
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K