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Topic: Countably many

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Countably infinite

	Countable set - Wikipedia, the free encyclopedia
		In mathematics the term countable is used to describe the size of a set, i.e.
		A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers.
		This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers by repeatedly mapping the first two elements to a natural number.
	en.wikipedia.org /wiki/Countable (1252 words)

	Countable set -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics the term countable set is used to describe the size of a (A group of things of the same kind that belong together and are so used) set, e.g.
		The notion of an infinite set is not elementary; it requires a strong sense of (A general concept formed by extracting common features from specific examples) abstraction and (The quality of being reproducible in amount or performance) precision, both from the beginning the distinguishing faculties of mathematicians.
		A set is called countable if the number of elements is finite or if it has the same number of elements as the (The number 1 and any other number obtained by adding 1 to it repeatedly) natural numbers.
	www.absoluteastronomy.com /encyclopedia/c/co/countable_set.htm (1270 words)

	Baire space - Wikipedia, the free encyclopedia
		Ignoring spaces with isolated points, which are their own interior, a Baire space is "large" in the sense that it cannot be constructed as a countable union of its points.
		Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.
		Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0, 1].
	en.wikipedia.org /wiki/Baire_space (732 words)

	Countable set
		A set is called countably infinite (or denumerable) if there exists a bijective mapping between it and the set N of all natural numbers.
		A countable set is a set which is either finite or countably infinite.
		A set which is not countable is uncountable.
	www.brainyencyclopedia.com /encyclopedia/c/co/countable_set.html (974 words)

	first category (Site not responding. Last check: 2007-11-05)
		Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.
		In particular, every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0,1].
		B is also of independent, but minor, interest in real analysis, where it is considered as a uniform space: the product of countably many copies of the discrete space N.
	www.yourencyclopedia.net /First_category.html (574 words)

	Countably many (Site not responding. Last check: 2007-11-05)
		A set is called countably infinite if there exists a bijectivemapping between it and the set N of all natural numbers.
		A countable (ordenumerable) set is a set which is either finite or countably infinite.
		A set which is not countable is called uncountable.
	www.therfcc.org /countably-many-176836.html (879 words)

	[No title]
		A set S is *countable* if it is the same size as the natural numbers N = {0,1,2,...}; that is, if there is a way to pair the elements of S with 0,1,2,...
		Countable sets are all the same size; their "size" is the size of N, countably infinite.
		For example, N is countable: take the identity pairing (n n) The set of integers Z = {...,-3,-2,-1,0,1,2,3,...} is countable: pair n with 2n, -n with 2n+1.
	www.cs.cornell.edu /html/cs212-fall98/lectures/l27-computability.txt (1724 words)

	COUNTABLY (Site not responding. Last check: 2007-11-05)
		"COUNTABLY" is a common misspelling or typo for: accountably, countable.
		In mathematics, a set is called finite if there exists a one to one correspondence, called a bijective mapping, between it and a set of the form {1, 2, 3,..., n} for some natural number n.
		A set is called countably infinite if there exists a bijective mapping between it and the set N of all natural numbers.
	www.websters-online-dictionary.org /definition/COUNTABLY (906 words)

	Baire space (Site not responding. Last check: 2007-11-05)
		Whenever the union of countably many closed subsets of X has aninterior point, then one of the closed subsets must have an interior point.
		Note that the space of rational numbers with the usual topologyinherited from the reals is not a Baire space, since it is the union ofcountably many closed sets without interior, the singletons.
		In particular, every non-empty Baire space is of second category in itself, and every intersection ofcountably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topologicaldisjoint sum of the rationals and the unit interval [0,1].
	www.therfcc.org /baire-space-191783.html (522 words)

	Peter Suber, "The Löwenheim-Skolem Theorem"
		We may add countably many proper axioms (axioms which are not logically valid wffs) to supplement the logical axioms (axioms which are logically valid wffs) of predicate logic.
		There are at most countably many strings of symbols (when the strings are finite in length).
		One countable model that is always available for inspection, if only to demystify LST a bit, is the interpretation in which the terms of the language are assigned to their own tokens, or to the typographic strings which express them.
	www.earlham.edu /~peters/courses/logsys/low-skol.htm (3032 words)

	First and Second Countable (Site not responding. Last check: 2007-11-05)
		You need to be familiar with the term countable, before we proceed.
		This is an indispensable property of first countable, and it is used in various proofs.
		Restrict radii to rational values, and the balls centered at p are countable.
	www.mathreference.com /top,12cnt.html (490 words)

	Countable - Wikipedia
		We now define a set to be countable if it is either finite or the same size as N (the set of positive integers).
		So the above constitutes a proof that the set of even integers is countable.
		THEOREM 2: The union of countably many countable sets is countable.
	nostalgia.wikipedia.org /wiki/Countable (438 words)

	Lecture 27: Self-Reference, Computability, and Undecidability (Site not responding. Last check: 2007-11-05)
		Since there are only countably many programs and uncountably many functions, there must be some function that isn't programmable.
		Countable sets are all the same size; their "size" is the size of
		Many of these constructions are related to the notion of self-reference.
	www.cs.cornell.edu /Courses/cs312/2001FA/lecture/lec27.htm (2029 words)

	Preprints
		Vaught's conjecture is the statement that any theory in a countable language of first-order predicate calculus has only countably many, or continuously many, distinct countable models.
		However, many valuable and insightful suggestions have not yet been acted on, and may be carried through in future drafts.
		Non-hyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, co-authored with Chris Good and Brian Raines provides examples of solenoid-like spaces (specifically, inverse limits of intervals under tent-maps) of uncountably many homeomorphism classes, using countable well-founded trees.
	www.maths.ox.ac.uk /~knight/stuff/preprints.html (510 words)

	Baire space (Site not responding. Last check: 2007-11-05)
		Note that the space of rational numbers with the usual topology inherited from reals is not a Baire space since is the union of countably many closed without interior the singletons.
		In particular every non-empty Baire space of second category in itself and every of countably many dense open subsets of X is non-empty but the converse of of these is true as is shown the topological disjoint sum of the rationals the unit interval [0 1].
		B is also of independent but minor in real analysis where it is considered as a uniform space : the product of countably many copies the discrete space N.
	www.freeglossary.com /First_category (768 words)

	[No title]
		The proof below is similar to the usual construction of continuum many almost-disjoint subsets of w as branches through a countably branching tree.
		I believe the style of arguments producing continuum many almost-disjoint subsets of omega or producing the claim above are at their limits above.
		Note that the mapping f is one-to-one, and its range consists of subsets of R of cardinality at most w_1; hence A N(x) is one-to-one and onto (hence, there are R* many N(x)'s), and (iii) N(x) has finite intersection with N(y) whenever x is not equal to y.
	www.math.niu.edu /~rusin/known-math/99/set_card (444 words)

	Loewenheim Skolem Theorem (Site not responding. Last check: 2007-11-05)
		In mathematical logic, the classic Löwenheim-Skolem theorem states that any "model" M has a countably infinite submodel N that satisfies exactly the same set of first-order sentences that M satisfies.
		Since there may be many such values of y, the axiom of choice must be invoked in order to infer the existence of the Skolem function.
		That subset of the model is the submodel whose existence the theorem asserts.
	www.wikiverse.org /loewenheim-skolem-theorem (291 words)

	Abelian Group Theory papers of Andreas R. Blass
		Noebeling proved that the subgroup B consisting of the bounded sequences is free and therefore has many homomorphisms to Z. We prove that all "reasonable" homomorphisms from B to Z factor through finite subproducts.
		We study, in the context of torsion-free abelian groups G, the sets that are maximal with respect to the property of freely generating a pure subgroup of G. We generalize many but not all of the familiar properties of basic subgroups to the subgroups generated by these "maximal pure independent" sets.
		Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable.
	www.math.lsa.umich.edu /~ablass/abgp.html (1114 words)

	RUSSELL'S ATTIC - Definition (Site not responding. Last check: 2007-11-05)
		a pair for each natural number), and countably many pairs of socks.
		Answer: countably many (map the left shoes to even numbers and the right shoes to odd numbers, say).
		Also countably many, we want to say, but we can't prove it without the Axiom of Choice, because in each pair, the socks are indistinguishable (there's no such thing as a left sock).
	www.hyperdictionary.com /computing/russell's+attic (93 words)

	Countable Set (Site not responding. Last check: 2007-11-05)
		This duplicitous behavior violated the customary international laws of war set forth in the 1907 Hague Convention on the Opening of Hostilities to which the...
		Schools need to be prepared, because in many cases budgets were set last January...
		A countable (or denumerable) set is a set which is either finite or countably infinite.
	www.wikiverse.org /countable-set (1026 words)

	Countable set - Wikipedia, the free encyclopedia
		THEOREM: The Cartesian product of finitely many countable sets is countable.
		What about infinite subsets of countably infinite sets?
		There are only countably many finite sequences, so also there are only countably many finite subsets.
	www.wikipedia.org /wiki/Countable (1252 words)

	Baire space
		Definition: A Baire space is a topological space X that satisfies one (and therefore all) of the following equivalent conditions: # Every intersection of countably many dense open sets is dense.
		# The interior of every union of countably many nowhere dense sets is empty.
		# Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point.
	en.mcfly.org /Baire_space (527 words)

	Godel's Theorems
		The set of subsets of N is isomorphic to the set of 0-1 sequences via the bijection between subsets and characteristic functions.
		There are uncountably many reals since the map which sends a 0-1 sequence 10101010...
		There are only countably many finite sequences of symbols and so there are only countably many programs and hence only countably many computable sequences.
	www.math.hawaii.edu /~dale/godel/godel.html (2115 words)

	Doug's Expositions (Site not responding. Last check: 2007-11-05)
		Given the axiom of choice, the union of countably many sets each of which is countable.
		The difference between countable and uncountable collections is quite important in topology and measure theory/probability.
		In measure theory, one axiom is that the measure is countably sub-additive: the measure of the union of countably many sets is at most equal to the sum of the measures.
	www.math.columbia.edu /~zare/cardinality.html (835 words)

	[No title]
		B(E)xB(E) is generated by rectangles A x B, and any one set in B(E)xB(E) belongs to the sigma-algebra generated by countably many rectangles.
		If otoh you're taking the modern "intersection of all the sigma-algebras containing" as the definition of "the sigma-algebra generated by" then the key hint is to show that the union of the sigma-algebras generated by the countable subsets of S is a sigma-algebra.) Now back to your question.
		Hence there's some countable collection of rectangles C = {(A_n)x(B_n)} so that the diagonal is in the sigma-algebra generated by C. But that's impossible if the cardinality of E is larger than c: In that case there must exist x and y with x
	www.math.niu.edu /~rusin/known-math/99/sigalg_prod (735 words)

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