Is a Bounded Set in R Countable if it Can be Covered by an Epsilon Cover?

  • Thread starter Thread starter sayan2009
  • Start date Start date
  • Tags Tags
    Bounded Set
Click For Summary
SUMMARY

The discussion centers on proving that a bounded subset A of the real numbers R is countable if it can be covered by an epsilon cover consisting of countable elements. The approach involves utilizing the properties of totally bounded sets and compactness of closed intervals. Specifically, the solution suggests covering A with open intervals that intersect A countably and leveraging the compactness of a closed interval containing A to establish the countability of A.

PREREQUISITES
  • Understanding of bounded sets in real analysis
  • Familiarity with the concept of epsilon covers
  • Knowledge of compactness in topology
  • Basic principles of countability in set theory
NEXT STEPS
  • Study the properties of totally bounded sets in real analysis
  • Learn about epsilon-delta definitions and their applications
  • Research compactness in metric spaces and its implications
  • Explore countability and its various forms in set theory
USEFUL FOR

Mathematicians, students studying real analysis, and anyone interested in set theory and topology will benefit from this discussion.

sayan2009
Messages
14
Reaction score
0

Homework Statement



given a set A(subset of R(reals)) is bounded.and for all x belongs to R there exists epsilon(eps) such that {(x-eps,x+eps) intersection A} is countable..to prove A is countable

Homework Equations





The Attempt at a Solution

...bdd set in R is totally bounded...but iam not finding the way how to cover A by epsilon cover(has at most countable elements)
 
Physics news on Phys.org
Take a closed interval C containing A and cover it with open intervals having a countable intersection with A. Now use the compactness of the closed interval C.
 
oooooooooooo gr888888888...thanks
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Image of a f with a local minima at all points is countable.
  • · Replies 16 ·
Replies
16
Views
2K
How should I show that all solutions of this equation are bounded?
  • · Replies 4 ·
Replies
4
Views
2K
Outer Measure is Countably Subadditive
  • · Replies 1 ·
Replies
1
Views
2K
A subset in R^n is bounded if and only if it is totally bounded.
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad The set of nonzero values of pmf is at most countable
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Basic question on 'bounded implies totally bounded'
  • · Replies 16 ·
Replies
16
Views
2K
Countably Compact: Necessity & Sufficiency
  • · Replies 1 ·
Replies
1
Views
2K
Prove that x is in S or x is an accumulation point of S
  • · Replies 25 ·
Replies
25
Views
3K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
3K