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.
(This page was last modified on 11 December 2016, at 18:54.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.)
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
(WordPress is blog for sharing)
— Copyright etc. —
Readers are welcome to copy, link to, quote from, or translate reasonable portions of the content of this blog (e.g. a single article) into other media, though for items longer than one or two paragraphs, I would appreciate it if a reference or citation to the URL that the content originates from is provided. If you wish to copy a significantly larger fraction of the content (e.g. an entire series of articles), please contact me about it first.
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.
(By posting your answer, you agree to the privacy policy and terms of service. site design / logo © 2017 Stack Exchange Inc; user contributions licensed under cc by-sa 3.0 with attribution required)
(definition of attribution: Licensees may copy, distribute, display and perform the work and make derivative works and remixes based on it only if they give the author or licensor the credits (attribution) in the manner specified by these.)
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)| ≤ Mx, ∀ 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 ∞, ∩ En≠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=∩ En≠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 En →0, then the intersection:
E=∩ En
(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)
- [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.
- [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.