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Topic: Separation axioms

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	Separation axiom - Wikipedia, the free encyclopedia
		The separation axioms are denoted with the letter "T" after the German "Trennung", which means separation.
		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.
	en.wikipedia.org /wiki/Separation_axiom (1489 words)

	PlanetMath: separation axioms
		The separation axioms are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.
		This is version 19 of separation axioms, born on 2003-02-23, modified 2005-07-02.
		This entry on separation axioms use the first one.
	planetmath.org /encyclopedia/SeparationAxioms.html (502 words)

	FBF - Biblical Viewpoint
		Separation is thought by some to be the mere "packaging" of Fundamentalism—the brown bag of a stale lunch leftover from an earlier day.
		Marriage is one of the simplest illustrations of separation, and even the most pragmatic opponent of the doctrine cannot defend his compromise against it.
		We’re not talking about separating from a disobedient brother—that is clearly Scriptural—but about cutting off the discussion as soon as a disagreement occurs with the assumption that the brother knows everything you know but has chosen to do wrong anyway.
	www.f-b-f.org /WebMan/Article.asp?ID=4126&Count=true (713 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Note that we sometimes use the terminology of separated sets to refer to points; in that situation, we're really talking about the singleton set {x} rather than the point x.
		Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms.
		In fact, in a normal space, any two disjoint sets will also be separated by a function; this is Urysohn's Lemma.
	www.online-encyclopedia.info /encyclopedia/s/se/separation_axiom.html (1450 words)

	Tychonoff space - Wikipedia, the free encyclopedia
		X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F.
		In fancier terms, this condition says that x and F can be separated by a function.
		(The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separation axioms.
	en.wikipedia.org /wiki/Tychonoff_space (600 words)

	Atlas: Generalized closed sets and weak separation axioms by Julian Dontchev (Site not responding. Last check: 2007-11-06)
		Separation axioms stand among the most common and to a certain extent the most important and interesting concepts in Topology.
		Most weak separation axioms are defined in terms of generalized closed sets.
		This inclines to indicate that further knowledge of the behavior of topological spaces satisfying these two weak separation axioms (and some related ones) is required.
	atlas-conferences.com /c/a/b/c/19.htm (317 words)

	Hausdorff and Other Separation Axioms for Topological Spaces (Site not responding. Last check: 2007-11-06)
		The fully general, abstract topological spaces (S,T), where S is a set and T is a collection of subsets of S that is closed under arbitrary unions and finite intersections and includes both the null set ∅ and the whole set S, is of limited interest.
		(Hausdorff Space Axiom): For any two points p and q in the topological space (S,T) there is a pair of disjoint open sets, one containing {p} and not containgin {q} and the other containing {q} and not {p}.
		(Regular Space Axiom): For any closed set C in the topological space (S,T) and any point p not in C there is a pair of disjoint open sets, one containing C and one containing {p}.
	www.applet-magic.com /separation.htm (285 words)

	2 The axiomatic approach (Site not responding. Last check: 2007-11-06)
		In order to avoid the many choices for the axiom of parallels we look at those axioms that are satisfied by a convex region of space (see Figure 1) in any axiomatic geometry--Euclidean or not.
		The first three axioms are the axioms of incidence and the latter two are the axioms of dimension for Projective geometry of 3 dimensions.
		The first three are the axioms of separation for Projective geometry and the last is a form of the Archimedean least upper bound principle.
	www.imsc.ernet.in /~kapil/papers/krp/node2.html (1709 words)

	Separation Axioms (Site not responding. Last check: 2007-11-06)
		These criteria are sometimes called the separation axioms, although this is a misnomer.
		If y is not in o then x and y are separable, and if y is in o then y and z are separable.
		These to open sets are disjoint, hence we have separated g and h.
	www.mathreference.com /top,sep.html (483 words)

	AXIOMS
		Any one of these axioms together with the axioms of Neutral geometry (incidence axioms I-1, I-2, I-3; betweenness axioms B-1, B-2, B-3, B-4; congruence axioms C-1, C-2, C-3, C-4, C-5, C-6; and Dedekind's continuity axiom) give a complete system of axioms for the Euclidean geometry.
		Perspectivities preserve separation; i.e., if (A, BC, D), with l the line through A, B, C, and D, and P, Q, R, and S are the corresponding points on line m under a perspectivity, then (P, QR, S).
		* in elliptic geometry the betweenness axioms are replaced with separation axioms.
	www.southernct.edu /~pinciuv/m360axio.html (960 words)

	ProvenMath (everything proven from axioms) - Apronus.com
		This page states the axioms of set theory in a formal and precise manner in terms of quantifiers, logical operators, and the membership symbol.
		The Axiom of Choice is given on a separate page where it is shown to be equivalent with the Well-Ordering Principle, Hausdorff's Maximal Principle, and Zorn's Lemma.
		We don't need the Axiom of Choice or the Axiom of Regularity to obtain this set but we use the Axiom of Regularity to show that the Natural Infinite Set is an ordinal and that all of its elements are ordinals.
	www.apronus.com /provenmath (985 words)

	James Tauber : Poincare Project: Separation Axioms
		It may seem an arbitrary restriction to go from a T_1 to a T_2 but it turns out that this additional requirement is what allows us to define a metric on a space or take unique limits of sequences.
		The additional axioms defined for T_0, T_1 and T_2 spaces say something about how separated the points have to be.
		For this reason they are referred to as separation axioms.
	jtauber.com /blog/2005/01/27/poincare_project:_separation_axioms (464 words)

	Axioms of Separation (Site not responding. Last check: 2007-11-06)
		Scripture teaches that believers and unbelievers cannot be yoked together in spiritual endeavor.
		Separation is not the answer to every disagreement between brethren.
		At any given time of church history, God is most severe on those whom he is using at the moment.
	www.obf.net /obf/axioms.htm (200 words)

	Normal space (Site not responding. Last check: 2007-11-06)
		X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint.
		In fancier terms, this condition says that E and F can be separated by neighbourhoods.
		That is, given disjoint closed sets E and F, there is a continuous function f from X to the real line R such the preimages of {0} and {1} under f are E and F respectively.
	usapedia.com /n/normal-space.html (914 words)

	Dr. William J. Pervin's Publications
		Separation axioms for syntopogenous spaces, Proceedings of the Royal Nederlands Academy of Sciences, Series A, vol.
		Separation axioms and metric-like functions, Pacific Journal of Mathematics, vol.
		On the separation axioms for bitopological spaces, Annales of the Social Sciences Society of Brussels, vol.
	www.utdallas.edu /~pervin/publications.html (471 words)

	AMCA: Some separation axioms in topological inverse semigroups by Peter Kortesi (Site not responding. Last check: 2007-11-06)
		The aim of this presentation is to study the separation axioms between T
		The given conditions show the importance of the set of idempotents for the separation of inverse semigroups.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/a/o/72.htm (179 words)

	Separation axioms
		This axiom is one of a number of separation axioms: T
		Note that the distance between disjoint closed sets may be 0 (but they can still be separated by open sets).
		By demanding more separation axioms one gets closer to a metric space.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L18.html (481 words)

	Separation Axioms (Site not responding. Last check: 2007-11-06)
		The 'cofinite' topology on an infinite set (where the open sets are those with finite complement) is T
		space if for every point a and closed set B there exist disjoint open sets which separately contain a and B. That is, points and closed sets are separated.
		space if for every pair of closed sets A and B there exist disjoint open sets which separately contain A and B. That is, points and closed sets are separated.
	www.math.toronto.edu /jjchew/math/topology/separation.html (355 words)

	Higher Separation Axioms (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: The hierarchy of separation axioms that is familiar from topological spaces generalizes to spaces with an isotone and expansive closure function.
		Neither additivity nor idempotence of the closure function must be assumed.
		4 On separation in topological space (context) - Sanin - 1943
	citeseer.ist.psu.edu /623117.html (419 words)

	Mathematics 746-747: Topology I,II (Site not responding. Last check: 2007-11-06)
		Topological spaces, convergence and continuity, separation axioms, compactness, connectedness, metrizability, complete metric spaces, homotopy, uniform spaces, and selected advanced topics.
		Basic definitions, neighborhoods, bases and subbases, subspaces, product spaces, weak and quotient spaces, separable spaces, countability properties.
		Separation by open sets, separation axioms and Hausdorff spaces, regular, completely regular and Tychonoff spaces, normal spaces.
	www.math.ndsu.nodak.edu /courses/746.html (147 words)

	Prof. Dr. M. Hosny Ghanim (Site not responding. Last check: 2007-11-06)
		A., A-compacness and separation axioms in fuzzy topological spaces, Bull.
		Mashhour, A. S., Ghanim, M. and Fathalla, M. A., a -separation axioms and a -compactness in fuzzy topological spaces, Rocky Mountain J. Math.
		Ghanim, M. H., Kerre, E. E and Mashhour, A. S separation axioms, subspaces and sums in fuzzy topological spaces J.
	math.vanderbilt.edu /~schectex/temp/topol/47.htm (728 words)

	General Topology (2301631) (Site not responding. Last check: 2007-11-06)
		Countability and Separation Axioms : Countability Axioms, Separation Axioms, Urysohn Lemma and Tietze Extension Theorem.
		Connectedness : Separations, Connectedness and Path-Connectedness, Components and Path Components, Local Connectedness and Local Path-Connectedness.
		Compactness : Covering, Compactness, Finite Intersection Condition, Tychonoff Theorem, Limit Point Compactness and Sequentially Compactness, Locally Compactness, Compactifications, One-Point Compactifications, Imbedding Theorem and Stone-Cech Compactification.
	www.math.sc.chula.ac.th /~phichet/topology (144 words)

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