Why Are There No Interior Points in the Set of Real Numbers?

Click For Summary
SUMMARY

The discussion centers on the mathematical assertion that the set of rational numbers (Q) has no interior points, nor does its complement, the set of irrational numbers (R\Q). An interior point requires the existence of a neighborhood around a point that consists entirely of members from the same set. Since any interval around a rational number contains irrational numbers, and vice versa, neither set can contain interior points. This conclusion is supported by the density of real numbers in both sets.

PREREQUISITES
  • Understanding of real numbers and rational numbers
  • Familiarity with the concept of interior points in topology
  • Knowledge of neighborhoods in metric spaces
  • Basic principles of set theory
NEXT STEPS
  • Study the concept of interior points in topology
  • Learn about metric spaces and neighborhoods
  • Explore the properties of dense sets in real analysis
  • Investigate the implications of the Cantor-Bernstein-Schröder theorem
USEFUL FOR

Mathematics students, educators, and anyone interested in topology and real analysis will benefit from this discussion.

omri3012
Messages
60
Reaction score
0
Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri
 
Physics news on Phys.org
Let q be an arbitrary rational number. Does there exist a neighborhood of q that is a subset of Q?
 
How about the fact that I can squeeze a real number between any two arbitrary points of Q?
 
trambolin said:
How about the fact that I can squeeze a real number between any two arbitrary points of Q?
Irrelevant. Do you mean an irrational number? Now that would be relevant.
 
omri3012 said:
Hallo,

My teacher wrote that:

"The set has no interior points, and neither does its complement, R\Q" where R refers real

numbers and Q is the rationals numbers.

why can't i find an iterior point?

thanks,

Omri
So "the set" is Q. For p to be an interior point of Q, there must exist an interval around p, [math](p-\delta, p+\delta)[/quote] consisting entirely of rational numbers. For p to be an interior point of R\Q, the set of irrational numbers, there must exist an interval (p- \delta, p+ \delta)] consisting entirely of irrational numbers. There is NO interval of real numbers consisting entirely of rational number or entirely of irrational numbers.
 
Last edited by a moderator:
That was what I said anyway, but of course a real is not necessarily rational part got lost along the way... Sorry for that.
 
thank you for your comments,

I'm sorry but the statement (as i guess you already assume) was:

"The set Q has no interior points, and neither does its complement, R\Q"

thanks

Omri
 
Yes, that was essentially what everyone was assuming.
 

Similar threads

Undergrad Algebraic property of real numbers
  • · Replies 30 ·
2
Replies
30
Views
3K
Graduate How to take the spatial derivative of quaternions
  • · Replies 8 ·
Replies
8
Views
3K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
6K
Insights What Are Numbers? - Insights for Beginners
  • · Replies 0 ·
Replies
0
Views
3K
Graduate Apostol definition of interior point and open set
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad On (real) entire functions and the identity theorem
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Are Real Numbers Essential in Scientific Measurements and Models?
  • · Replies 85 ·
3
Replies
85
Views
9K
Undergrad On a proof of the converse of the supporting hyperplane theorem
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K