Cantor's theorem - Factbites

Cantor's theorem - Factbites
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Topic: Cantor's theorem

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	Theorem
		Van der Waerden's theorem Van der Waerden's theorem is a theorem of the branch of integers.
		Theorem of Bolzano-Weierstrass The theorem of Bolzano-Weierstrass in convergent subsequence.
		Fundamental theorem of Riemannian geometry In metric tensor.
	www.brainyencyclopedia.com /topics/theorem.html

	Cantor set - free-definition
		Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge.
		Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space.
		The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers.
	www.free-definition.com /Cantor-set.html

	Topology - Encyclopedia.WorldSearch
		Georg Cantor, the inventor of set theory, had begun to study the theory of point sets in Euclidean space, in the later part of the 19th century.
		Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth".
		The Tietze extension theorem : In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
	encyclopedia.worldsearch.com /topology.htm

	Georg Cantor - Wikipedia, the free encyclopedia
		Cantor is also known for his work on the unique representations of functions by means of trigonometric series (a generalized version of a Fourier series).
		Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm.
		Today, the vast majority of mathematicians accept Cantor's work on transfinite sets and recognize it as a
	www.wikipedia.org /wiki/Georg_Cantor

	Axiomatic set theory - Wikipedia, the free encyclopedia
		Cantor's development of set theory was still "na�ve" in the sense that he didn't have a precise
		Cantor gave two proofs that R is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had manifold applications in logic and mathematics.
		Cantor went right ahead and constructed infinite hierarchies of infinite sets, the
	en.wikipedia.org /wiki/Axiomatic_set_theory

	PlanetMath: Heine-Cantor theorem
		This is version 6 of Heine-Cantor theorem, born on 2002-06-07, modified 2003-03-17.
		I believe R can be replaced by an arbitrary uniform space in this theorem.
	planetmath.org /encyclopedia/HeineCantorTheorem.html

	PlanetMath: proof of Heine-Cantor theorem
		This is version 2 of proof of Heine-Cantor theorem, born on 2003-03-17, modified 2003-03-21.
		"proof of Heine-Cantor theorem" is owned by paolini.
		Cross-references: continuous, subsequences, convergent, compact, sequences, uniformly continuous, metric spaces, theorem
	planetmath.org /encyclopedia/ProofOfHeineCantorTheorem.html

	Set Theory: Cantor
		By the end of the nineteenth century Cantor was aware of the paradoxes one could encounter in his set theory, e.g., the set of everything thinkable leads to contradictions, as well as the set of all cardinals and the set of all ordinals.
		Cantor first was able to drop the condition that the coefficients be Fourier coefficients -- consequently any trigonometric series convergent on an interval had coefficients converging to 0.
		We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather because he initiated the study of infinite sets and numbers which have provided such fascinating material, and difficulties, for logicians.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/cantor/cantor.html

	Search Results for Cantor
		I Grattan-Guinness, The rediscovery of the Cantor -Dedekind correspondence, Jahresberichte der Deutschen Mathematiker-Vereinigung 76 (1974), 104-139.
		Cantor also discussed the concept of dimension and stressed the fact that his correspondence between the interval [0, 1] and the unit square was not a continuous map.
		Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Cantor&CONTEXT=1

	Heine [Definition]
		Alice Heine Alice Heine (February 10, 1858 - December 22, 1925), was the American born wife of Prince Albert I of Monaco, a great-grandfather of Prince Rainier III of Monaco....
		Cincinnatus Heine Miller (= Joaquin Miller Joaquin Miller was the penname of the hyperbolical American eccentric Cincinnatus Heine (or Hiner) Miller (September 8, 1837, or November 10, 1841 - February 17, 1913).
		Eduard Heine Heinrich Eduard Heine (March 15, 1821 in Berlin - October 21, 1881 in Halle (Saale)) was a German mathematician....
	www.wikimirror.com /Heine

	Heine-Cantor theorem - Wikipedia, the free encyclopedia
		Eduard Heine and Georg Cantor, states that if M is a compact metric space, then every continuous function
	en.wikipedia.org /wiki/Heine-Cantor_theorem

	The Ultimate List of theorems Dog Breeds Information Guide and Reference
		In some fields, theorem can be considered as a courtesy title, given to major results, although with a content that would not satisfy a mathematician.
		No attempt is made here to comment on that aspect of usage: this is a list of results known as theorems.
		Abel-Ruffini theorem ( theory of equations, Galois theory)
	www.dogluvers.com /dog_breeds/List_of_theorems

	List of mathematical proofs
		Articles devoted to theorems of which a (sketch of a) proof is given
		Theorems of which articles are exclusively devoted to proving them
		2 Articles devoted to theorems of which a (sketch of a) proof is given
	www.objectssearch.com /encyclopedia/en/wikipedia/l/li/list_of_mathematical_proofs.html

	PlanetMath: uniformly continuous
		Any continuous function defined on a compact space is uniformly continuous (see Heine-Cantor theorem).
		Cross-references: Heine-Cantor theorem, compact, sequences, uniform convergence, Cauchy sequences, property, distances, uniform spaces, metric spaces, uniform, domain, function, continuous, positive, real, subset, real function
		Uniformly continuous functions have the property that they map Cauchy sequences to Cauchy sequences and that they preserve uniform convergence of sequences of functions.
	planetmath.org /encyclopedia/UniformlyContinuous.html

	Re: [HM] Did Cantor prove the Schroeder-Bernstein theorem? by Carlos Cesar de Araujo
		I mentioned one text where is said that our "polemical" theorem was "proved" by Cantor, whereas almost everyone call it "Schroeder-Bernstein theorem".
		I know Banach-Knaster-Whittaker-Tarski's fixed-point theorem very well and how to derive "Cantor-Bernstein" theorem from it as well as many other interesting things (maximal principles, Banach-Tarski "paradox" etc.).
		Then I went on to give a reason, and the reason is: Cantor's proof relies essentially on the comparability between arbitrary cardinals, while Bernstein's proof does not.
	mathforum.org /epigone/historia/sminsmooplum/71w3qkfh4hu8@forum.swarthmore.edu

	PlanetMath: proof of Schroeder-Bernstein theorem
		proof of Cantor-Bernstein theorem, proof of Cantor-Schroeder-Bernstein theorem
		Cross-references: cardinality, map, composition, theorem, identity map, disjoint, counterexample, minimal, pairwise disjoint, subsets, sequence, bijection, injection, lemma
		This is version 14 of proof of Schroeder-Bernstein theorem, born on 2002-07-05, modified 2004-02-14.
	planetmath.org /encyclopedia/ProofOfCantorSchroederBernsteinTheorem.html

	Cardinal number
		Cantor invented the one-to-one correspondence, which easily showed that two finite sets had the same cardinality if there was a one-to-one correspondence between the members of the set.
		But, later this year, Cantor succeeded in proving that there were higher-order cardinal numbers using the ingenious but simple Cantor's diagonal argument.
		Cantor also developed a lot of the general theory of cardinal numbers; he proved that there is a cardinal number that is the smallest (
	www.worldhistory.com /wiki/C/Cardinal-number.htm

	Encyclopedia: Heine-Cantor theorem
		In mathematics, the Heine-Cantor theorem states that if M is a compact metric space, then every continuous function
		PlanetMath: Heine-Cantor theorem ( http://planetmath.org/?op=getobjandfrom=objectsandid=3066) (proof at [1] ( http://planetmath.org/?op=getobjandfrom=objectsandid=4114))
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Heine_Cantor-theorem

	Biography of Kronecker
		Cantor, Lindemann, Heine, Dedekind, and Weierstrass were just some of the mathematicians who held different beliefs on math than Kronecker.
		Not only did Kronecker feel that Dedekind, Heine and Cantor's views on mathematics were an unacceptable way of thinking, but he also created some friction between himself and Weierstrass.
		So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments.
	www.andrews.edu /~calkins/math/biograph/biokrone.htm

	Weierstrass, Dedekind and cantor
		Cantor was a friend and colleague of Dedekind, but he had a far different life.
		Among these, for example, belongs the above-mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis.
		In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences.
	www.maths.uwa.edu.au /~schultz/3M3/L28Weierstr,Dede,Cantor.html

	DEEC
		The Riemann integral: integrability conditions, integrability of monotone functions and continuous functions; mean-value theorem, indefinite integral, the fundamental theorem of calculus, integration by parts and by substitution, apllication to calculus of line lengths and areas of regions in the plane.
		Curves and line integrals: arc length, integrals relative to arc length, line integrals, the fundamental theorems of calculus for line integrals, conservative fields and scalar potentials, application to the principle of conservation of mechanical energy.
		Taylor's theorem, applications to the study of maxima and minima.
	wwwgire.ist.utl.pt /intercambio/Acordos_Socrates_en/ECTS/ECTS-DEEC.html

	Mathematics Courses
		Cauchy theorem and its applications, calculus of residues, expansions of analytic functions, analytic continuation, conformal mapping and Riemann mapping theorem, harmonic functions.
		Possible topics are: congruences, reciprocity laws, quadratic forms, prime number theorem, Riemann zeta function, Fermat’s conjecture, diophantine equations, Gaussian sums, algebraic integers, unique factorization into prime ideals in algebraic number fields, class number, units, splitting of prime ideals in extensions, quadratic and cyclotomic fields, partitions.
		Further topics, selected by instructor, such as exterior differential forms, Stokes’ theorem, manifolds, Sard’s theorem, elements of differential topology, singularities of maps, catastrophes, further topics in differential geometry, topics in geometry of physics.
	www.ucsd.edu /catalog/courses/MATH.html

	Encyclopedia: Uniformly continuous function
		If M is a compact metric space, then every continuous f : M → N is uniformly continuous (this is the Heine-Cantor theorem).
		Every Lipschitz continuous map between two metric spaces is uniformly continuous.
		This function is continuous, but not uniformly continuous, since as x approaches 0, the changes in f ( x) grow beyond any bound.
	www.nationmaster.com /encyclopedia/Uniformly-continuous-function

	Differential and integral calculus 1
		Bolzano's theorem, Weierstrass theorem, uniform continuity and Cantor's theorem.
		Consequences of completeness : Dedekind's theorem and its equivalence to the completeness, R is Archimedian ordered, the set Q of rationals is dense in R, the nested intervals property of R, the existence of n-th roots for positive real numbers.
		Heine's definition for limits and its equivalence to Cauchy's definition.
	www.tau.ac.il /~leviatan/calc1.html

	Syllabus query -- 2002/2003
		Global continuity, mean value theorem, Weierstrass and Heine-Cantor theorems.
		Countable and uncountable subsets of R. Sequences: notion of convergence and fundamental theorems; infinite limits and indeterminations.
		Understanding the necessity of the Supremum Axiom and its implications in the fundamental results given in the course; understanding the non-finite character of the limit concept; recognizing the scope of Lagrange theorem.
	lci.math.ist.utl.pt /prog.phtml.en?disc=AMI&sem=1&ano=1&anolectivo=2002

	IMACS
		An introduction to equipollence and dominance in preparation for the study in "Elements of Mathematics" Book 8 of cardinal numbers; Cantor's Theorem; the Schrder-Bernstein Theorem; the Cantor discontinuum.
		Differentiability; the Linear Approximation Theorem; properties of derivatives; the calculus within its historical context; the Mean Value Theorem for Derivatives; curve sketching; the chain rule; parametric representation of relations; various forms of l'Hpital's rule; Cauchy's Mean Value Theorem; implicit differentiation; antiderivatives.
		The axiom of choice; the Hausdorff Chain Theorem; Zorn's lemma; the Well-Ordering Theorem; the principle of transfinite induction; Bourbaki's Theorem; transfinite recursion; ordinal numbers; cardinal numbers; a discussion of the continuum hypothesis; the Fundamental Theorem of Cardinal Arithmetic.
	www.imacs.org /IMACSWeb?page=Mathematics

	Automated Theorem Proving
		Gentzen's (Hauptsatz) Theorem (and the subformula property) is tttp.html:with sequents)--deriving theorems from "not A or A" (with A atomic) tttp.html:theorems).
		In fact, any tttp.html:Tarski's Theorem is proven (that truth in a model M is not definable vonHeijenoort.html:(more or less) proved his theorem giving countable models, which was vonHeijenoort.html:hinting at Godel's Incompleteness Theorems.
		This provides a confirmation of Godel's theorem and szabo.html:does not believe Godel's theorem is surprising and is a relatively szabo.html:("Main Theorem"-- cut-elimination) which implies that we can put every szabo.html:examples of theorems in predicate calculus.
	www.andrew.cmu.edu /user/cebrown/notes/thmtemp

	Cantor-Bernstein-Schroeder theorem
		The theorem is also known as the Schroeder-Bernstein Theorem, but the trend is towards adding Cantor's name to properly credit him.
		An earlier proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem.
		In set theory, the Cantor-Bernstein-Schroeder theorem is the theorem that if there exist injective functions
	www.sciencedaily.com /encyclopedia/cantor_bernstein_schroeder_theorem

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