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Topic: Axiom of countability

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Topological space

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	Axiom - Wikipedia, the free encyclopedia
		In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results.
		Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups).
		The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four.
	en.wikipedia.org /wiki/Axiom (1598 words)

	Axiom (Site not responding. Last check: 2007-10-22)
		The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any universe, they will always hold no matter what values are assigned to the variables.
		Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups).
	hallencyclopedia.com /Axiom (1498 words)

	Axiom - the free encyclopedia (Site not responding. Last check: 2007-10-22)
		The word axiom comes from the Greek wordαξιωμα (axioma), which means that which is deemed worthy or fit or that which is consideredself-evident.
		Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in anyuniverse, they will always hold no matter what values are assigned to the variables.
		Reasoning about two differentstructures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim tocapture what is special about a particular structure (or set of structures, such as algebraic groups).
	www.encyclopedia-of-knowledge.com /?t=Axiom (1203 words)

	Axioms of countability (Site not responding. Last check: 2007-10-22)
		Axioms of countability are two properties satisfied by some topological spaces.
		We say that a topological space satisfies the first axiom of countability if it has a countable base of neighbourhoods at every point.
		Obviously the second axiom of countability implies the first one; the converse is in general not true.
	www.uncover.us /en/wikipedia/a/ax/axioms_of_countability.html (136 words)

	axiom (Site not responding. Last check: 2007-10-22)
		As the word axiom is understood in modern mathematics, an axiom is not a proposition that is self-evident.
		For example, in some rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate.
		An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
	www.yourencyclopedia.net /axiom.html (756 words)

	PlanetMath: first axiom of countability
		A space that satisfies the first axiom of countability is said to be first-countable.
		"first axiom of countability" is owned by drini.
		This is version 2 of first axiom of countability, born on 2002-02-18, modified 2004-03-29.
	planetmath.org /encyclopedia/FirstAxiomOfCountability.html (66 words)

	Axiom of countability - Wikipedia, the free encyclopedia
		In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
		separable spaces: there exists a countable dense subspace,
		Lindelöf spaces: every open cover has a countable subcover,
	en.wikipedia.org /wiki/Axiom_of_countability (140 words)

	PlanetMath: second countable
		A topological space is said to be second countable if it has a countable basis.
		It can be shown that a second countable space is both Lindelöf and separable, although the converses fail.
		This is version 12 of second countable, born on 2002-01-01, modified 2005-02-11.
	planetmath.org /encyclopedia/CompletelySeparable.html (119 words)

	First-countable space - Wikpedia (Site not responding. Last check: 2007-10-22)
		In topology, a first-countable space is a topological space satisfying the "first axiom of countability".
		An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).
		Any countable product of a first-countable space is first-countable, although uncountable products need not be.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=First-countable_space (232 words)

	The word axiom comes from the Greek Greek word alpha xi...
		The word "axiom" comes from the Greek Greek word αξιωμα ("axioma"), which means that which is deemed worthy or fit or that which is considered self-evident self-evident.
		As the word "axiom" is understood in modern mathematics mathematics, an axiom is "not" a proposition that is self-evident.
		An axiom is an elementary basis for a formal logic logic system that together with the rules of inference define a logic.
	www.biodatabase.de /axiom (620 words)

	Axiom - Art History Online Reference and Guide (Site not responding. Last check: 2007-10-22)
		In mathematics, an axiom is not a self-evident truth but rather, a formal logical sentence that constitutes a fundamental brick in the development of a theory.
		In all its formalism, the Peano axioms constitute the most widely used axiomatization of arithmetic; these are a set of axioms strong enough to prove several relevant facts of number theory and they allowed Gödel to establish his famous second incompleteness theorem.
		This collection of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23).
	www.arthistoryclub.com /art_history/Axiom (1609 words)

	axiom information (Site not responding. Last check: 2007-10-22)
		The word axiom comes from the Greek wordαξιωμα (axioma), which means that which is deemed worthy or fit or that which isconsidered self-evident.
		Logical axioms, as the mere formulas that they are, are void of any meaning; but the point is that when they becomeinterpreted in any given universe, they will always hold no matter what values are assigned to the variables.
		Thus, this notionof axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter when, where orwhy.
	www.vsearchmedia.com /axiom.html (1234 words)

	separation axiom
		The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points.
		The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be separated in some stronger sense.
		Most of these axioms have alternative definitions with the same meaning; the definitions given here are those which fall into a consistent pattern relating the various notions of separation defined in the previous section.
	www.fact-library.com /separation_axiom.html (1463 words)

	iqexpand.com (Site not responding. Last check: 2007-10-22)
		This might challenge the classical notion of axiom and is at least one of the reasons why axioms are not regarded as obviously true or self-evident statements.
		Axiom loudspeaker prototypes undergo rigorous measurement and psychoacoustical testing at the National Research Council (NRC).
		Axiom Celebrates the Michel-Novelozo Gallery Saturday, January 08, 2005 - 07:55 AM (263 Reads) Axiom invites you to visit the Michel-Novelozo Gallery of Photography, located on the ground-floor of our...
	axiom.iqexpand.com (1611 words)

	List of axioms . Zermelo set theory . Boolean prime ideal theorem . Wightman axioms (Site not responding. Last check: 2007-10-22)
		Axiom of countability topology Gluing axiom sheaf theory Haag-Kastler axioms quantum field theory Huzita s axioms origami Kuratowski closure axioms topology Peano s axioms natural numbers Probability axioms Separation axiom topology Wightman axioms quantum field theory
		The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily...
		Basically, the idea of the Wightman axioms is there is a Hilbert space upon which the Poincaré group acts unitary representation unitarily.
	www.uk.fraquisanto.net /List_of_axioms (203 words)

	Second-countable space explained (Site not responding. Last check: 2007-10-22)
		In topology, a second-countable space is a topological space satisfying the "second axiom of countability".
		Like other countability axioms, the property of being second-countable restricts the number open sets that a space can have.
		Although, the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates.
	www.wordspider.net /se/second-countable-space.html (611 words)

	Axiom - Open Encyclopedia (Site not responding. Last check: 2007-10-22)
		For the algebra software named Axiom, see Axiom (algebra software).
		This axiom simply states that if we know $\forall x P(x)\, for some property P\,, and t\, is particular term in the language (i.e., it stands for a particular object in our structure), then we should be able to claim P(t)\, .$
		The formal issue arises in the need to derive what logicians call a deductive system, which consists of a set $\Lambda of logical$ axioms, a set $\Sigma of non-logical$ axioms and a set $\{(\Gamma, \phi)\} of rules of inference.$
	open-encyclopedia.com /Axiom (1357 words)

	Axiom (Site not responding. Last check: 2007-10-22)
		In the propositional calculus it is common to take as logical axioms all formulas of the following forms, where $\phi, \psi, and \chi can be any formulas of the language:$
		A deductive system consists of a set $\Lambda\, of logical$ axioms, a set $\Sigma\, of non-logical$ axioms, and a set $\{(\Gamma, \phi)\}\, of rules of inference.$
		Axiom is also the name of a 3D graphics engine.
	www.apawn.com /search.php?title=Axiom (1656 words)

	The first axiom of countability (from topology) -- Encyclopædia Britannica (Site not responding. Last check: 2007-10-22)
		This axiom is modeled after the fact that if p is any point in the plane and U is an open set containing p, then there is an integer n such that the open ball with centre at p and radius 1/n lies in U.
		A topological space X satisfies the first axiom of countability provided the following condition holds: for each point p in X there is a sequence of open sets U
		More results on "The first axiom of countability (from topology)" when you join.
	www.britannica.com /eb/article-69121 (785 words)

	ONT Re: Topology
		Since the second axiom of countability has been mentioned, it seems only
		countability is therefore definitely more restrictive than the first.
		Hence the existence of a countable local subbase at each point implies the first axiom
	suo.ieee.org /ontology/msg03903.html (295 words)

	Axiom - Gurupedia
		philosophers of the ancient Greeks an axiom was a claim which could be seen to be true without any need for proof.
		As the word axiom is understood in modern
		mathematics, an axiom is not a proposition that is self-evident.
	www.gurupedia.com /a/ax/axiom.htm (655 words)

	axiom : QuicklyFind Info (Site not responding. Last check: 2007-10-22)
		Category:Algebra The word 'axiom' comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		For instance, (misquoting Peano) simple arithmetic including addition can be defined and many theorems proven by assuming # a number called 0 exists # every number X has a successor called inc(X) # X+0 = X # inc(X) + Y = X + inc(Y)
		Using these axioms, and defining the customary short names 1, 2, 3, and so on for inc(0), inc(inc(0)), inc(inc(inc(0))) respectively, we can show that:
	www.quicklyfind.com /info/axiom.htm (690 words)

	The second axiom of countability (from topology) -- Encyclopædia Britannica (Site not responding. Last check: 2007-10-22)
		A topological space satisfies the second axiom of countability if it has a countable basis—that is, interiors of circles with radii and centres represented by rational numbers.
		More results on "The second axiom of countability (from topology)" when you join.
		More from Britannica on "The second axiom of countability (from topology)"...
	www.britannica.com /eb/article-69122 (831 words)

	DC MetaData for: A better framework for first countable spaces (Site not responding. Last check: 2007-10-22)
		Abstract: In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients.
		Several applications of first countability in a broader context than the usual one of topological spaces are studied.
		Keywords: First axiom of countability, second axiom of countability, countably compact, sequentially compact, sequentially complete, continous convergence, sequentially continous, semiuniform convergence spaces, convergence spaces, filter spaces, topological spaces, uniform spaces, bicoreflective subconstruct, cartesian closedness
	www.math.fu-berlin.de /publ/preprints/2002/Ab-A-02-01.html (138 words)

	Axiom bei eLexi - das Onlinelexikon (Site not responding. Last check: 2007-10-22)
		\nThe word axiom comes from the Greek word\nαξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident.
		For example, in some\nrings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate.
		\n* Axiomatic system\n* Peano axioms\n* Axiom of choice\n* Axiom of countability\n* Axiomatic set theory\n* Parallel postulate\n* Continuum hypothesis\n* Axiomatization
	www.elexi.de /en/a/ax/axiom.html (905 words)

	Atlas: First countability and the Axiom of Choice by Gonçalo Gutierres (Site not responding. Last check: 2007-10-22)
		The definition of first countable space is well established, although its (usual) formulation highly relies on the Axiom of Choice.
		In order to obtain, at same time, countable neighborhood bases to every point of the space we have to make arbitrary choices.
		Starting from this observation, we will see alternative definitions of first countability that turn out to be non-equivalent, and we study in what conditions their equivalence remains true.
	atlas-conferences.com /cgi-bin/abstract/cajp-62 (169 words)

	The first axiom of countability (from topology) -- Britannica Concise Encyclopedia - The online encyclopedia you can ...
		To get another Hausdorff space not satisfying this first axiom of countability the topology of the line
		The basis elements are defined to be of two sorts: any one point set other than the origin, or the complement of a finite set of points.
		Then the first axiom of countability is not satisfied at the origin.
	www.britannica.com /ebc/article-69121 (886 words)

	Axioms of countability camelsbtoes Axioms of countability (Site not responding. Last check: 2007-10-22)
		Find axioms of countability and more at Lycos Search.
		axioms of countability in the Free Online Encyclopedia
		Read about axioms of countability in the free online encyclopedia and dictionary.
	www.find-ask.com /Encyclopedia/Axioms_of_countability/Axioms_of_countability.html (168 words)

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