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	PlanetMath: cofinite and cocountable topology
		together with the cofinite topology forms a compact topological space.
		"cofinite and cocountable topology" is owned by saforres.
		This is version 12 of cofinite and cocountable topology, born on 2002-09-17, modified 2005-10-08.
	planetmath.org /encyclopedia/Cofinite.html (82 words)

	Reference.com/Encyclopedia/Cofinite
		In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set.
		The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation.
		In this case, the non-principal ultrafilter is the set of all cofinite sets.
	www.reference.com /browse/wiki/Cofinite (298 words)

	Cofinite - Wikipedia, the free encyclopedia
		The cofinite topology (sometimes called the finite complement topology) is a topology which can be defined on every set X.
		The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology.
		An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together.
	en.wikipedia.org /wiki/Cofinite_topology (532 words)

	[No title]
		Cofiniteness is critical because many argumen* *ts and constructions proceed inductively.
		Y be a cofinite directed level representation of a * *map in a pro-category.
		Proof.It suffices to assume that there is a cofinite directed level representat* ion for the diagram f h g X ____//_Yoo__Z _____//W in which f and g are levelwise weak equivalences while h is a pro-isomorphism ( *but not a levelwise isomorphism).
	hopf.math.purdue.edu /IsaksenD/strict.txt (6131 words)

	Cofinite: Encyclopedia topic (Site not responding. Last check: 2007-10-29)
		In mathematics (mathematics: A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), a cofinite subset of a set X is a subset Y whose complement (complement: Number needed to make up whole force) in X is a finite set.
		The set of all subsets of X that are either finite or cofinite forms a Boolean algebra (Boolean algebra: A system of symbolic logic devised by George Boole; used in computers), i.e., it is closed under the operations of union, intersection, and complementation.
		One place where this concept occurs naturally is in the context of the Zariski topology (Zariski topology: in mathematics, the zariski topology is a structure basic to algebraic geometry,...
	www.absoluteastronomy.com /reference/cofinite (467 words)

	Encyclopedia: Compact space (Site not responding. Last check: 2007-10-29)
		Slightly more generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
		Any space carrying the cofinite topology is compact.
		(In the cofinite topology, a set is open iff it is empty or its complement is finite.)
	www.nationmaster.com /encyclopedia/Compact-space (3691 words)

	Citebase - Graded cofinite rings of differential operators
		Proof: Since A is cofinite in D(X), its 0-component A is cofinite in the 0-component O(X) of D(X).
		Since A ⊆ Dc (B1, A) ∩ Dc (B2, A) is cofinite in D c (B) we conclude ˆˆ ˆ that D c (B2, A) is cofinite in D c (B1).
		Proof: Clearly, K is cofinite in L. This implies that IL ⊆ L is a proper ideal whenever I ⊂ K is a proper ideal.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0403493 (4803 words)

	Cofinite
		In mathematics, a cofinite subset of a set X is a subset Y such that its complement in X is a finite set.
		Put another way, the open sets in this topology are the cofinite sets, plus the empty set.
		Brought to you by TravelSources and the Beaches and Towns Network, LLC.
	www.teachtime.com /en/wikipedia/c/co/cofinite.html (175 words)

	Boolean algebra - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-29)
		The smallest element 0 is the empty set and the largest element 1 is the set S itself.
		The set of all subsets of S that are either finite or cofinite is a Boolean algebra.
		For any natural number n, the set of all positive divisors of n forms a distributive lattice if we write a ≤ b for a divides b.
	encyclopedia.worldsearch.com /boolean_algebra.htm (1551 words)

	Fall 2000 - Math 225A - Final - Russell O’Connor
		Cofinite sets are done in a similar manner, except that σ(i) = ¬[∀x(ψ
		Suppose that R defines a subset of ω; that is neither finite nor cofinite.
		The set of even numbers is neither finite nor cofinite.
	r6.ca /math225/final.html.txt (627 words)

	Does Python need a '>>>' operator? (Site not responding. Last check: 2007-10-29)
		> ^^^ 2 is missing > > is cofinite because its complement, which is {2}, is finite.
		If you did extend long integers (interpreted as finite and cofinite bitstrings) to bitstrings that are neither finite nor cofinite, you would end up with an interesting mathematical set known as the 2-adic integers.
		Two 2-adic numbers are close to each other if their difference is divisible by a high power of 2.
	mail.python.org /pipermail/python-list/2002-April/098914.html (324 words)

	CMPSCI 601 Q&A for HW#5, Spring 2003
		This is certainly cofinite but not of the form you gave.
		Every cofinite set must contain all numbers above some number c, but then it may also include some or all of the numbers less than c as well.
		Yes, it is true that the language COFINITE is properly contained in INFINITE, and that you have given an example of a number/machine that is in INFINITE but not in COFINITE.
	www.cs.umass.edu /~barring/cs601s03/qahw5.html (1163 words)

	[No title]
		The correspondence between endofinite objects and cofinite ideals is the main theme* * of Sec- tion 3 which contains the proofs of Theorem A and Theorem B. In the following t* wo sections we apply the results from the previous ones to the stable homotopy cat *egory of spectra.
		Conversely, if Ann X is cofinite, t* hen FX (A) is a finite length X -module for every A in A. It follows from [13, Theorem 9.6 ] and the characterization of pure-injectives in [14, Theorem 1.8] that X is pure-injecti *ve.
		Thus I is a maximal cofinite cohomological ideal in C0; in particular X = EI.
	hopf.math.purdue.edu /KrauseH-Reichenbach/endofiniteness.txt (6691 words)

	ANSWERS TO TESTS (Site not responding. Last check: 2007-10-29)
		Any collection S of subsets of X is a subbasis for a topology on X. The real numbers with the cofinite topology (other examples exist).
		To show that f(x)=x^2 is continuous with respect to the cofinite topology, it is enough to show that if C is a closed set then f^(-1)(C) is closed.
		But the cofinite topology on X x Y has no infinite proper closed set.
	spot.colorado.edu /~kearnes/S04/topt.html (442 words)

	Solution 5
		Thus the usual topology has more closed subsets thatn the cofinite topology and hence more open subsets.
		Hence every finite subset is closed and so every finite set is open in the subspace topology which is therefore discrete.
		In the cofinite topology on R, the subset 2Z is not closed and hence it is not closed in the subspace topology on Z.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Solutions/S5.html (315 words)

	T1 space: Encyclopedia topic (Site not responding. Last check: 2007-10-29)
		Every cofinite (cofinite: in mathematics, a cofinite subset of a set x is a subset y whose complement in...
		However, this example is well known as a space that is not Hausdorff (Hausdorff: in topology and related branches of mathematics, a hausdorff space, separated...
		For a more concrete example, let's look at the cofinite topology (cofinite topology: in mathematics, a cofinite subset of a set x is a subset y whose complement in...
	www.absoluteastronomy.com /reference/t1_space (1323 words)

	Math Forum: Ask Dr. Math: A Mathematical Essay
		Since this is a cofinite set, it is in a nonprincipal ultrafilter and it is large.
		Next, consider the set M of all proper filters that are generated by H. One of the elements in M, for example, could be the set of all cofinite sets, plus all even integers, plus all supersets and intersections.
		No element in M is principal, because since an element B of M contains all cofinite sets, B cannot contain one-element sets, for then B would contain the empty set (the intersection of one-element set and the cofinite complement), and this would contradict the fact that B is proper.
	mathforum.org /dr.math/faq/analysis_hyperreals.html (9021 words)

	[No title]
		A directed set I is cofinite if for every t, the set of elements of I less th* *an t is finite.
		The cofiniteness is critical because many construct* *ions and proofs proceed inductively.
		W indexed by a cofinite directed set I for wh* *ich there is a strictly increasing function n : I !
	hopf.math.purdue.edu /IsaksenD/prospace.txt (6765 words)

	Abstract for Strongly Zeroed Distance Spaces (Site not responding. Last check: 2007-10-29)
		In this paper, we construct an example of a countable strongly zeroed distance space whose induced topology is cofinite.
		We then show that no uncountable cofinite space can be induced by a strongly zeroed distance.
		We conclude that the property of being stringly zeroed is not likely to be highly correlated to topological properties.
	www.math.uprm.edu /~hajek/cv/strnzerabs.htm (66 words)

	Abstracts of my papers
		--- On cofinite subgroups of mapping class groups, Proceedings of 9th Gokova Geometry-Topology Conference, Turkish Journal of Mathematics 27 No.
		For every positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism onto a free group of rank $n$.
		We conclude that $H^1(\Gamma_n;\Z)$ has rank at least $n$ and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum.
	arf.math.metu.edu.tr /~korkmaz/abstracts.html (1346 words)

	Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe. (Site not responding. Last check: 2007-10-29)
		Vinberg has given a lemma relating $\z$-lattices of signature $(n,1)$ having a cofinite reflection group with certain positive definite, reflective sublattices, i.e.
		As a consequence of these results and Vinberg's lemma, the maximal dimension of $\z$-lattices of signature (n,1) with a cofinite reflection group is determined to be n+1 = 22.
		Key words and phrases: Lorentzian lattice, rootsystem, reflective lattice, reflexive lattice, cofinite reflection group, arithmetic reflection group.
	www.matha.mathematik.uni-dortmund.de /preprints/96-01.html (131 words)

	Discrete and Indiscrete Topologies (Site not responding. Last check: 2007-10-29)
		In the indiscrete topology, only the empty set and the entire set are open and closed.
		In the cofinite topology, all finite sets are closed, along with the entire set (as required).
		If the entire set is finite, the cofinite topology gives the discrete topology.
	www.mathreference.com /top,disc.html (143 words)

	The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite ...
		The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups, Feraydoun Taherkhani
		The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups
		We show that the mapping class group of a closed surface of genus 2 does not satisfy the Kazhdan property by constructing subgroups of finite index having a nonvanishing first cohomology group.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1045952350 (117 words)

	Topology/Topological Spaces - Wikibooks, collection of open-content textbooks
		is a topology on X called the cofinite topology on X.
		If x is in X, then a subset N is called a neighborhood of x if there is an open set U such that
		Show that the cofinite (respectively, cocountable) topology on a set X equals the discrete topology if and only if X is finite (respectively, countable).
	en.wikibooks.org /wiki/Topological_Spaces (592 words)

	Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory (Springer Monographs in Mathematics)
		Starting off with several models of hyperbolic space and its group of motions the authors discuss the spectral theory of the Laplacian and Selberg's theory for cofinite groups.
		The interplay with arithmetic is demonstrated by means of the groups PSL(2) over rings and of quadratic integers, their Eisenstein series and their associated Hermitian forms.
		A rich chapter on concrete examples of arithmetic and non-arithmetic cofinite groups enhances the usefulness of this work for a wide circle of mathematicians.
	www.uni-protokolle.de /buecher/isbn/3540627456 (245 words)

	T1 space
		However, this example is well known as a space that is not Hausdorff (T
		For a more concrete example, let's look at the cofinite topology on an infinite set.
		Specifically, let X be the set of integers, and define the open sets O
	www.brainyencyclopedia.com /encyclopedia/t/t1/t1_space.html (852 words)

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