Compactness

Bridget Dunbar

https://en.wikipedia.org/wiki/Compact_space

I think this website will be a good resource because of the diagrams shown to help explain. The sections are laid out in a simple manner that give the reader a chance to understand the basic concepts first that will be used to define compactness and its properties.

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https://terrytao.wordpress.com/2009/02/09/245b-notes-10-compactness-in-topological-spaces/

This website has a good layout. It describes the topic gives definitions and important facts, then gives examples and other lemmas/ corollaries that are important

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http://math.stackexchange.com/questions/371928/what-should-be-the-intuition-when-working-with-compactness

I think this website will be helpful because its people asking questions on parts of the topic they are confused on and others answering those questions.

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I do not think I will be able to use this one  ^^^^^^ 

Additional Sources I could possibly use if the above two are not enough

http://www.mathcs.org/analysis/reals/topo/compact.html

MathCS.org – Real Analysis: 5.2. Compact and Perfect Sets

Exercises :

Elementary Real Analysis, Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner.

Prentice Hall, 2001, xv 735 pp.

page 172 number 4.5.10, 4.5.14, 4.5.15

page 173 number  4.5.16, 4.5.17, 4.5.21

Bridget Dunbar

December 15, 2017

Objective 3X Part 1

Compactness Arguments

The idea of compactness looks at the set as a whole to relate the ideas of closed and open sets.

Local Boundedness looks at the “local” assumptions. Suppose a function, f , is locally bounded at each point in set E. Meaning, that every point x ∈E there exists an interval (x-δ, x δ) and f is bounded on the points in E that belong to (x-δ, x δ).

The idea of local boundedness is expanded to relate to the while function:

|f(t)| ≤ M_x, ∀ t ∈ E in the interval (x-δ, x δ)

Globally we want:

|f(t)| ≤ M, ∀ t ∈ E

These ideas result in the Local Boundedness Theorem: Suppose a function f is locally bounded at each point of a set E that is closed and bounded. Then f is bound with the whole set E.

Compactness Arguments detail the special characteristics of closed and bounded sets of real numbers.

Bolzano-Weierstrass Property: A set of real numbers, E, is closed and bounded if and only iff every sequence of points chosen from the set has a subsequence that is convergent to a point that is also an element in E.

Corollary: A set E is closed and bounded if and only if every infinite subset has an accumulation point belonging to E.

Bolzano-Weierstrass Theorem: Suppose a function f, is locally bounded at each point of a closed, bounded set E. Then f is bounded on the entire set E

Cantor’s Intersection Property: This involves the intersection of a descending sequence of sets where,

E₁ ⊃ E₂ ⊃ E₃ ⊃ E₄ ⊃_….

and given some conditions we have:

from i=1 to ∞, ∩ E_n≠0

Cantor’s Intersection Theorem: Let {Sn} be a sequence of real numbers which has closed and bounded subsets (also nonempty) such that:

E₁ ⊃ E₂ ⊃ E₃ ⊃ E₄ ⊃_….

Let E=∩ E_n≠0

(for intersection from i=1 to ∞)

Then E is not empty.

Corollary: Cantor’s Intersection Theorem: Suppose {En} that is a sequence of nonempty closed subsets of real numbers such that:

E₁ ⊃ E₂ ⊃ E₃ ⊃ E₄ ⊃_….

If diameter E_n →0, then the intersection:

E=∩ E_n

(for intersection from i=1 to ∞)

Consists of one point.

Cantor’s Intersection Theorem: Suppose a function, f, is locally bounded at each point of a set E that is closed and bounded. Then f is bound on the entire set E.

Exercises:

1) Compare completeness of the following intervals: and [0,1). (Wachsmuth)

1. [0,1] is compact. This can be seen by taking any sequence of points in the sequence. Since the sequence is bounded, by Bolzano-Weierstrass we have a convergent subsequence.
2. [0,1) is not compact. There exists subsequences that converge to 1, which is not a part of the set, thus the set is not compact. (Wachsmuth)

Works Cited

Thomas, Bruckner and Bruckner; Section 4.5

Wachsmuth, Bert, Interactive Real Analysis. 2017. Section 5.2.