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Topic: Cocountable topology

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Cofinite topology

List of examples in general topology

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	PlanetMath: countable complement topology
		is countable, then the countable complement topology is just the discrete topology, as the complement of any set is countable and thus open.
		Though defined similarly to the finite complement topology, the countable complement topology lacks many of the strong compactness properties of the finite complement topology.
		This is version 1 of countable complement topology, born on 2004-09-24.
	planetmath.org /encyclopedia/CountableComplementTopology.html (122 words)

	PlanetMath: cofinite and cocountable topologies
		together with the cofinite topology forms a compact topological space.
		"cofinite and cocountable topologies" is owned by yark.
		This is version 18 of cofinite and cocountable topologies, born on 2002-09-17, modified 2006-12-09.
	planetmath.org /encyclopedia/Cocountable.html (81 words)

	► » sequences and topologies (Site not responding. Last check: 2007-10-12)
		The indiscrete topology is the smallest topology for S assuring A. If T is bunch of topologies that assure A, then sup T assures A. if (aj,a) in A, a in U some subbase set of sup T, then
		There are two topologies for uncountable S that model this A. The discrete topology and the cocountable topology.
		A topology is a subset of P(S) A collection of topologies for a set S is a subset of P(P(S)).
	www.science-chat.org /sequences-and-topologies-3618128.html (1579 words)

	Von Neumann algebra (Site not responding. Last check: 2007-10-12)
		A von Neumann algebra or W-algebra (named for John von Neumann) is a -algebra of bounded operators on a Hilbert space that is closed in the weak operator topology, and contains the identity operator.
		In this definition the weak (operator) topology can be replaced by almost any other common topology other than the norm topology, in particular by the strong or ultrastrong topologies.
		Note that such algebras are rarely separable in the norm topology.
	www.toshare.info /en/W-star-algebra.htm (3849 words)

	Completely Hausdorff space
		In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space.
		Readers of textbooks in topology must be sure to check the definitions used by the author.
		The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology.
	music.musictnt.com /biography/sdmc_Urysohn_space (419 words)

	Topology Course Lecture Notes
		, the topology for X induced by the metric d, is defined by agreeing that G shall be declared as open whenever each x in G is contained in an open ball entirely in G, i.e.
		That form of definition is useless in the absence of a properly defined 'distance' function but, fortunately, it is equivalent to the demand that the preimage of each open subset of the target metric space shall be open in the domain.
		We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence).
	at.yorku.ca /i/a/a/b/23.dir/ch1.htm (2430 words)

	Cocountable - Wikipedia, the free encyclopedia
		If the complement is finite, then one says Y is cofinite.
		It is the smallest σ-algebra containing every singleton set.
		Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology.
	en.wikipedia.org /wiki/Cocountable (201 words)

	Topology Course Lecture Notes
		Sequences are of immense usefulness in real analysis and in metric spaces and elsewhere - and their failure to describe general topology adequately is a technical handicap.
		This topology is 'just right' in the sense that it is barely fine enough to guarantee the continuity of the coordinate projection functions while being just course enough allow the important result of Theorem.
		A basic formal distinction between algebra and topology is that although the inverse of a one-one, onto group homomorphism [etc!] is automatically a homomorphism again, the inverse of a one-one, onto continuous map can fail to be continuous.
	at.yorku.ca /i/a/a/b/23.dir/index.htm (8277 words)

	Maths Course 321 (Site not responding. Last check: 2007-10-12)
		This course assumes some prior exposure to Toplogy as found in course 212 and analysis as found in course 221.
		The course contains some more advanced topics in General Topology, with a small dose of fundamental topics at the beginning.
		Deals with metric and topological spaces: bases, neighbourhood bases, weak toplogies, cocountable topology on an uncountable set is not Hausdorff and yet no sequence has more than one limit.
	www.maths.tcd.ie /~mkerrin/Courses/321/old (360 words)

	Amazon.com: "dyadic rationals": Key Phrase page (Site not responding. Last check: 2007-10-12)
		Key Phrases in this book: Compare Problem, Reconsider Problem, cocountable sets, lim infan, normal number theorem, centered normal distribution, submartingale relative, simple random variables, separable with respect, fixed discontinuities, nth play, nonnegative case (See more)
		Key Phrases in this book: independence and product measures, conditional expectation and martingales, upper variation, regular conditional probability, monotone class theorem, subsequence that converges, absolutely continuous probability, finitely additive probability, integrable random variables, covering lemma, maximal function, elementary outcomes (See more)
		A General Topology Workbook by Iain T. Adamson
	www.amazon.com /phrase/dyadic-rationals (516 words)

	Amazon.com: "countable index": Key Phrase page (Site not responding. Last check: 2007-10-12)
		They need not be indexed by the integers; any countable index set will do.
		Schaum's Outline of General Topology by Seymour Lipschutz
		See all pages with references to countable index.
	www.amazon.com /phrase/countable-index (484 words)

	Completely Hausdorff space - Wikipedia, the free encyclopedia
		Let x and y be points in X.
		One can find counterexamples showing that none of these implications reverse
		This page was last modified 20:47, 24 November 2006.
	en.wikipedia.org /wiki/Completely_Hausdorff_space (459 words)

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