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Topic: Cofinite topology

Related Topics

Cocountable topology

Subspace (topology)

Topology glossary

List of examples in general topology

Compact space

Topological space

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	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/CofiniteTopology.html (81 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.
		The discrete topology for X is the finest one; the trivial topology is the coarsest.
		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)

	Topological space
		A purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
		Any infinite set with the cofinite topology (i.e., the open sets are the empty set and the sets whose complement is finite).
		Metric spaces were defined and investigated by Fréchet in 1906, Hausdorff spaces by Felix Hausdorff in 1914 and the current concept of topological space was described by Kuratowski in 1922.
	www.ebroadcast.com.au /lookup/encyclopedia/to/Topological_subspace.html (829 words)

	Cofinite - One Language (Site not responding. Last check: 2007-11-03)
		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.
		One place where this concept occurs naturally is in the context of the Zariski topology.
	www.onelang.com /encyclopedia/index.php/Cofinite_topology (314 words)

	WebCT test questions
		While the intersection of 2 topologies is a topology, it is not always true that the union of 2 topologies is a topology.
		In the cofinite topology on R (the same as the Zariski topology on R), [a,b) is neither open nor closed.
		Topology is a mind-stretching subject that frees us from conventional geometry and allows us to appreciate geometric characteristics of objects that remain unchanged even when the objects are stretched, shrunk, distorted, or contorted.
	www.mathsci.appstate.edu /~sjg/class/4710/tests/testquestions.html (3365 words)

	T1 space
		topology", with the term "Fréchet space" which refers to an entirely different notion from functional analysis.
		The Zariski topology on an algebraic variety is T
		For a more concrete example, let's look at the cofinite topology[?] on an infinite set.
	www.ebroadcast.com.au /lookup/encyclopedia/ac/Accessible_space.html (564 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).
	www.math.purdue.edu /research/atopology/IsaksenD/strict.txt (6131 words)

	Solution 5
		In the usual topology finite sets are closed, but there are some closed subsets that are not finite.
		In the cofinite topology on R, the subset 2Z is not closed and hence it is not closed in the subspace topology on Z.
		Topologies with the same number of sets are on the same level.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Solutions/S5.html (315 words)

	[No title]
		The topology P consists of the elements in O and the complements of closed subsets as neighborhoods of that point.
		Hausdorff topology +------------------------------------------------------------ The Hausdorff topology is a metric on the set of closed bounded subsets of a complete metric space.
		induced topology +------------------------------------------------------------ The induced topology on a subset A of X, where (X,T) is a topological spoace is the the topological space (A, Y cap A _Y in T).
	www.math.harvard.edu /~knill/sofia/data/topology.txt (1652 words)

	Topology
		Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
		Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular computational complexity theory.
		In topology a boundary component of a compact surface is a connected component consisting of boundary points of a surface.
	www.shortopedia.com /T/O/Topology (1036 words)

	Topology Reading Course
		Students are expected to become familiar with the basic concepts and methodology of point-set topology: separation properties, connectedness, and compactness, as well as subspaces, quotient spaces, and the properties of continuous mappings.
		Since the intersection of the open intervals (-1/2^n,1/2^n) is {0}, one should not expect a topology to be closed under infinite intersections.
		That is, excluding the trivial smallest topology (consisting of only the empty set and the entire set, called the ``co-discrete'' topology) and the trivial largest topology (which consists of _all_ subsets and is called the ``discrete'' topology), there are 27 nontrivial topologies.
	www.georgetown.edu /faculty/kainen/topol-02.html (1223 words)

	Discrete and Indiscrete Topologies (Site not responding. Last check: 2007-11-03)
		Given a space t, with its own topology, a set of points s embedded in t is discrete if each point in s is contained in an open set that contains none of the other points of s.
		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).
	www.mathreference.com /top,disc.html (143 words)

	m425
		The textbook is "Topology", 2nd edition, by T. Gamelin and R. Greene published by Dover ISBN 0-486-40680-6.
		Prove that a set X with the cofinite topology is compact.
		Prove that an infinite set X with the cofinite topology is not Hausdorff.
	ac.marywood.edu /johnsonc/www/m425.htm (770 words)

	Reference.com/Encyclopedia/T1 space
		The cofinite topology on an infinite set is a simple example of a topology that is T
		Then the basis of the topology are given by finite intersections of the subbasis sets: given a finite set A, the open sets of X are
		The Zariski topology is essentially an example of a cofinite topology.
	www.reference.com /browse/wiki/R0_space (944 words)

	Exercises 5
		Show that the subspace topology on any finite subset of R is the discrete topology.
		Show that the subspace topology on the subset Z is not discrete.
		Show that there are 29 different topologies on the set {a, b, c}.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T5.html (187 words)

	Topology Course Lecture Notes
		{ p }; it is cofinite and is thus an (open) neighbourhood of p.
		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.
		One confirms that d is a metric, and induces the original topology.
	at.yorku.ca /i/a/a/b/23.dir/ch5.htm (954 words)

	Zariski Topology
		The zariski topology is consistent with the metric topology, though it is weaker.
		has the cofinite topology, hence r is metric closed, but not zariski closed.
		Let k be a field with a valuation metric, such as the p-adic numbers, or the fraction field of any pid, or the completion thereof.
	www.mathreference.com /ag,zar.html (549 words)

	ANSWERS TO TESTS
		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)

	Cocountable at AllExperts
		The cocountable topology (sometimes called the countable complement topology) on any set X consists of the empty set and all cocountable subsets of X.
		In the cocountable topology, the only closed subsets are countable sets, or the whole of X.
		Then X is automatically LindelÃ¶f in this topology, since every open set only omits countably many points of X.
	en.allexperts.com /e/c/co/cocountable.htm (225 words)

	I like topology- what can I do with it?
		Sorry I'm really naive about all this, I've only taken a couple classes on it and I think it's cool, and I'm trying to figure out "what I want to do with my life," and maybe this is where to start asking questions.
		In any case, the subject is just an indispensable tool in many parts of mathematics and physics, though I seriously cannot think of a 'job' that uses topology explicitly.
		I don't tihnk you should choose your career based upon maths course preferences like that; it isn't like one single course will lead into a career; they are all inextricably linked.
	www.physicsforums.com /showthread.php?t=117011 (900 words)

	BSc Course Outline
		This course is concerned with the study of measure theory and integration.
		This course includes the basic operations of set theory (union, intersection etc.); mappings including subjective, injective, injective and inverse; countable and uncountable sets including the diagonal process and the unaccountability of R and countable unions of countable sets.
		Ideas of continuity, neighbourhood and open set; standard topologies on R and Rn and discrete, indiscrete and cofinite topologies on a general set; other topologies on R; closed sets; metrics in R and Rn, convergence of sequences in a metric space, Cauchy sequences and Completeness.
	web.lums.edu.pk /~webdev/BSc_Course_outline/BSc_courseoutline8.htm (1036 words)

	Topology Seminar (Site not responding. Last check: 2007-11-03)
		We will cover the basics of point-set topology, including simple applications in calculus, as well as some rudimentary aspects of differentiable topology, leading up to a brief treatment of the recent solution of the Poincare conjecture.
		A topology on set X is a collection tau of subsets of X such that: (i) tau contains the empty set and X (X is required to be nonempty) (ii) the intersection of any two members of tau is in tau (iii) the union of any subfamily of tau is in tau.
		#2 Show that the collection of all subsets of X forms a topology for any nonempty set X. Let the cofinite topology kappa on some infinite set X be defined to be the collection of all subsets of X which have a finite complement, together with the empty set.
	www.georgetown.edu /faculty/kainen/topsem.html (623 words)

	[No title]
		With this topology, show that there exists a continuous function defined on the union of $X$ and $Y$ that agrees with $f$ and $g$ on $X$ and $Y$, respectively.
		An example of a compact subset that is not closed (in a non-Hausdorff space) is a singleton in the indiscrete topology.
		Suppose ${\cal S}$ is a topology on $A$ relative to which $p$ is continuous.
	www.math.ucsb.edu /~moore/quals.txt (4939 words)

	Amazon.com: "relative topology": Key Phrase page (Site not responding. Last check: 2007-11-03)
		it is called the relative topology on A or the relativization of T to A, and the topological space (A, TA) is called a subspace of...
		The set S is nonempty and we shall now show that S is open and closed in the relative topology of V6(p) X V6(p).
		Then WA is a topology on A, called the relative topology on A. Key Phrases in this book: edge path group, glueing lemma, open simplices, covering homotopy theorem, common simplex, closed simplex, geodesic coordinate system, second structural equation, open simplex, first structural equations, simplicial map, parallel translate (See more)
	www.amazon.com /phrase/relative-topology (502 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		Pursuit of the Continuum Hypothesis has motivated a lot of useful and interesting mathematics in real analysis, topology, set theory, and logic.
		set of all cofinite sets of natural numbers (sets whose complements are finite)
		Example: R with the usual topology (open sets = open intervals) is ccc.
	www.ii.com /math/ch (4563 words)

	Separation Axioms (Site not responding. Last check: 2007-11-03)
		The 'particular point' topology (where the open sets are the sets containing a particular point a) is T
		The 'cofinite' topology on an infinite set (where the open sets are those with finite complement) is T
		The Zariski topology on a vector space (whose closed sets are the intersections of zeroes of polynomials on the coordinates of the vector space) is T
	www.math.toronto.edu /~jjchew/math/topology/separation.html (355 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.
	www.math.metu.edu.tr /~korkmaz/abstracts.html (1557 words)

	Topics in Topology: Rigidity theorems
		The geometry and topology of 3-manifolds, by W. Thurston, available at the MSRI website.
		Definition of the topology on the boundary at infinity.
		Reduction of the remaining gaps in establishing Step (2) to a single Key Fact: that in a delta hyperbolic geodesic space, every quasi-geodesic is at a uniformly bounded distance (in terms of the quasi-isometry constants) from a geodesic.
	www.math.ohio-state.edu /~jlafont/TopClass.html (1327 words)

	errata & addenda for TOPOLOGY AND UNIFORMITY (chapters 15-21)
		The proof asserts that the set S-sub-lambda is closed, but that's simply not true under the assumptions being used (i.e., using the cofinite topologies).
		To see that, note that each coordinate projection map is continuous (by the definition of product topology) hence measurable, and the product sigma-algebra is the smallest sigma-algebra which makes the coordinate projections all measurable.
		To see that, consider any open set in the product topology; it can be written as a union of countably many open rectangles.
	www.math.vanderbilt.edu /~schectex/ccc/addenda/partc.html (1946 words)

	Topology Course Lecture Notes
		For example, in (R,), (0,1) is not closed yet it is compact (since its topology is the cofinite topology!)
		Further, in a metric space, a closed bounded subset needn't be compact (e.g.
		(iii) In Q (with its usual topology), the components are the singletons.
	at.yorku.ca /i/a/a/b/23.dir/ch2.htm (2489 words)

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