Nested intervals, uncountable sets, unique points.

In summary, the problem is to prove that for an interval [a,b] and a subset A of [a,b], if A is uncountable, then there exists a point z in [a,b] such that the intersection of A with any open interval containing z is also uncountable. The solution may involve using the fact that a set containing an uncountable subset is also uncountable, and considering the completeness axioms/theorems such as the least upper bound, bounded monotone sequences, Bolzano-Weierstrass, Heine-Borel, nested intervals, limit points, accumulation points, or subsequences.
  • #1
mathkiddi
4
0

Homework Statement



Let [a,b] be an interval and let A be a subset of [a,b] and suppose that A is an infinite set.

Suppose that A is uncountable. Prove that there exists a point z which is an element of [a,b] such that A intersect I is uncountable for every open interval I that contains z.

Homework Equations



I don't really know how to start this problem. I know I can use the fact that a set that contains an uncountable subset is uncountable. Any help would be appreciated

The Attempt at a Solution



I know z is an element of I, and that I is uncountable. I also know z is an element of [an, bn]. I know there exists a unique point z in [an, bn] and I know A is uncountable.
 
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  • #2
Do you know about compactness?
 
  • #3
no, we have not covered compactness.
 
  • #4
mathkiddi, what "completeness" axioms/theorems have you covered? Specifically, have you covered any of these:

least upper bound, bounded monotone sequences, Bolzano-Weierstrass, Heine-Borel, nested intervals, or anything like these? limit points? accumulation points? subsequences?

I also know z is an element of [an, bn].

What is [an, bn]?
 

What are nested intervals?

Nested intervals refer to a mathematical concept where there is a sequence of intervals, where each interval contains the next one. This means that the intervals are nested within each other.

What are uncountable sets?

Uncountable sets are sets that have an infinite number of elements and cannot be counted or listed in a finite amount of time. This means that there are infinitely many elements in the set.

What is the significance of unique points in nested intervals?

In the context of nested intervals, unique points refer to the points that are common to all the nested intervals. These unique points are important because they help us to understand the behavior and properties of the nested intervals.

How are nested intervals and uncountable sets related?

Nested intervals and uncountable sets are related because uncountable sets often have nested intervals within them. This is because uncountable sets have infinitely many elements, and therefore can be divided into smaller and smaller intervals, creating a nested structure.

What are some real-world applications of nested intervals and uncountable sets?

Nested intervals and uncountable sets are used in a variety of fields, including mathematics, physics, and computer science. In mathematics, they are used to study the behavior of real numbers and to prove important theorems. In physics, they are used to model the motion of continuous systems. In computer science, they are used in algorithms for data compression and storage.

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