Does every uncountable set of reals contain an interval?

  • Context: Graduate 
  • Thread starter Thread starter dreamtheater
  • Start date Start date
  • Tags Tags
    Interval Set
Click For Summary

Discussion Overview

The discussion revolves around whether every uncountable subset of the real numbers must contain at least one interval, exploring various examples and counterexamples. The scope includes theoretical considerations and examples from set theory.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that uncountable sets of reals, such as the Cantor set, may not contain any intervals.
  • One participant suggests that if a subset of the reals contains no intervals, it might be denumerable, although they express uncertainty about this claim.
  • Another participant counters the initial claim by listing the Cantor set, the irrationals, and the transcendental numbers as counterexamples to the idea that uncountable sets must contain intervals.
  • A later reply elaborates on the Cantor set, arguing that it can be shown to contain no intervals of any length by considering the construction of the set.

Areas of Agreement / Disagreement

Participants express disagreement regarding whether uncountable sets must contain intervals, with multiple competing views presented, particularly concerning the Cantor set and other examples.

Contextual Notes

Limitations include the need for further clarification on the definitions of intervals and the properties of the sets discussed, as well as unresolved mathematical reasoning regarding the Cantor set's construction.

dreamtheater
Messages
10
Reaction score
0
Let S be an uncountable subset of the reals.

Then does S always contain at least one interval (whether it be open/closed/half-open/rays/etc..)?

Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?

My intuition is that, if any subset of the reals contains no intervals, then it must be denumerable, but this might be wrong.
 
Physics news on Phys.org
dreamtheater said:
Let S be an uncountable subset of the reals.

Then does S always contain at least one interval (whether it be open/closed/half-open/rays/etc..)?

Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?

My intuition is that, if any subset of the reals contains no intervals, then it must be denumerable, but this might be wrong.

Consider the set of all irrational numbers.
 
False!
The Cantor, the irrationals, and the transendential numbers are obvious counter examples.
 
dreamtheater said:
Maybe the Cantor set is an example of an uncountable set that contains no intervals? How does one show this if it is true?
Take any interval. It has length d. But for sufficiently high n, 3^{-n} < d. Therefore, by construction, the Cantor set contains no intervals of length d.
 

Similar threads

High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad An open set can be a closed interval?
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Is the Set of Integer Outputs of sin(x) Sequentially Compact in ℝ?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Name for a subset of real space being nowhere a manifold with boundary
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad On a proof of the converse of the supporting hyperplane theorem
  • · Replies 9 ·
Replies
9
Views
3K
MHB Discrete sets and uncountability of limit points
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Co-finite topology on an infinite set
  • · Replies 2 ·
Replies
2
Views
4K