Compact-open topology

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Topic: Compact-open topology

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	Compact-open topology - Wikipedia, the free encyclopedia
		The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis.
		In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.
		If X is compact, and if Y is a metric space with metric d, then the compact-open topology on C(X, Y) is metrisable, and a metric for it is given by e(f, g) = sup{d(f(x), g(x)) : x in X}, for f, g in C(X, Y).
	en.wikipedia.org /wiki/Compact-open_topology (412 words)

	Compactly generated space - Wikipedia, the free encyclopedia
		is the space of continuous maps from X to Y with the compact-open topology.
		In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X.
		For instance, every locally compact (and compact) space is compactly generated, as is every first-countable space.
	en.wikipedia.org /wiki/Compactly_generated_space (427 words)

	compact.opn
		The importance is that it is what is called an admissible topology for continuous functions, and for locally compact spaces, the weakest one.
		Consider a compact set in the topology of the argument, and any neighborhood of the range of that compact set.
		Then those functions which map the compact set into that neighborhood form a neighborhood in the c-o topology.
	www.math.niu.edu /~rusin/known-math/93_back/compact.opn (242 words)

	PlanetMath: compact-open topology
		This is version 5 of compact-open topology, born on 2003-02-05, modified 2003-02-07.
		Cross-references: converges, sequence, compact sets, uniform convergence, metric space, uniform space, subbasis, generated by, open set, subspace, compact, continuous maps, topological spaces
		is a metric space), then this is the topology of uniform convergence on compact sets.
	planetmath.org /encyclopedia/CompactOpenTopology.html (89 words)

	General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk)
		Compact spaces are very important in general topology for a variety of reasons.
		If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces.
		This is indeed a topology, called the quotient topology induced on Y by
	noisefactory.co.uk /maths/topology.html (4788 words)

	Current Seminars
		Compact monothetic subesemigroups of S(R) have been characterized by Boyce in 1971, and Magill asked in 1991 if a reasonable description of finite monothetic subsemigroups could be given for this semigroup.
		We provide partial answers to the question ``Is the original topology of A the only complete norm topology making all the operators from A continuous?", for multipliers and other operators.
		APRIL 22: Javier Trigos California State University, Bakersfield, "Non-measurable subgroups of compact metric abelian groups".
	home.att.net /~topann/FebApr2004.html (1353 words)

	diploma_main.txt
		The compact open topology is the same as the topology of uniform convergence on compact subsets [Dugundji, Theorem XII 7.2].
		En passant, it will be proved that the compact open topology and the C1-topology (the smooth counterpart of the compact open topology) on the isometry groups coincide.
		The compact deter- mination sets play in some way the r^ole of the compactness of the standard simplex (as can be seen in Lemma (3.45)) and the finiteness of singular chains in singular homology.
	hopf.math.purdue.edu /Strohm/diploma_main.txt (17691 words)

	PlanetMath: $\mathcal{C}^r$ topologies
		compact-open topology, is defined in the same fashion but instead of choosing
		Whitney (or strong) topology is a topology assigned to the space
		is compact, the weak and strong topologies coincide.
	planetmath.org /encyclopedia/CompactOpenMathcalCrTopology.html (191 words)

	ab-6.htm
		It has been observed by T. Edwards that the compact-open topology on the family of continuous functions between topological spaces is the weak topology induced on the family by a certain collection of continuous compact-valued functions.
		In this paper, several of Edwards’ results are generalized to the F-open topology of A. Wilansky, and it is shown that Edwards’ discovery leads to a compact-open topology for families of set-valued functions between topological spaces.
		In addition, several theorems on the first countability of the space of compact-valued functions with one of the compact-open topologies of Smithson are proved.
	www.pphmj.com /abstracts/jpgt/vol4issue1/ab-6.htm (193 words)

	mat530 weekly
		A sub-basis for this topology is indexed by the pairs (K,U) where K is compact in X and U open in Y.
		Proposition: if X is locally compact, and F: X x I --> Y is a continuous map, then the map f: I --> C(X,Y) defined by ft(x) = F(x,t) is a continuous map with respect to the compact-open topology.
		Proof that compactness (defined in terms of open covers) is equivalent to the Finite Intersection Property.
	www.math.sunysb.edu /~tony/archive/top/weekly.html (4044 words)

	CMS/CSHPM Summer 2005 Meeting
		In particular, let X be a compact ordered space, Y a Hausdorff space, and let F(Y) denote the family of all nonempty closed subsets of Y with the Vietoris topology.
		A connected open subset U of the sphere is called pseudo convex if for all points z in U there exist at most two closest points in the boundary of U.
		In particular, every boundary-less 3-manifold admits a flow with a discrete set of fixed points and such that the closure of every non-trivial trajectory is 2-wild, which answers a question posed at the 2004 Spring Topology and Dynamics Conference.
	www.cms.math.ca /Events/summer05/abs/gta.html (2548 words)

	Algebraic Topology: Topology
		A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open.
		The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous.
		The topology on B defined by F is the strongest topology (i.e., the largest collection OB) for which all these functions become continuous.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	Topologies on Spaces of Continuous Functions (Abstract)
		Continuity of the function-evaluation map is shown to coincide with a certain approximation property of a topology on the frame of open sets of the exponent space, and the existence of a smallest approximating topology is equivalent to exponentiability of the space.
		The only prerequisite to this development is a basic knowledge of general topology (continuous functions, product topology and compactness).
		We show that the intersection of the approximating topologies of any preframe is the Scott topology.
	rw4.cs.uni-sb.de /~heckmann/abstracts/topfunc.html (236 words)

	PlanetMath:
		compactness is preserved under a continuous map owned by yark
		proof of characterization of connected compact metric spaces.
		closed set in a compact space is compact owned by mathcam
	planetmath.org /encyclopedia/C (3742 words)

	Margie Hale's Education
		In the compact-open topology, H(M) is regular, Lindelof, paracompact, F-stable, and non-first countable.
		The dissertation deals with the topology of some direct limit spaces and their related homeomorphism groups.
		Topology, Homeomorphisms of Some Direct Limit Spaces, directed by Richard E. Heisey.
	www.stetson.edu /~mhale/vita/educat.htm (254 words)

	Abstracts.html
		It is proved that the automorphism group of a locally compact connected double loop is a locally compact transformation group with respect to the compact-open topology.
		For arbitrary compact connected projective planes P of finite dimension n M. Lueneburg derived upper bounds for the dimension of G depending on n and the configuration of the fixed elements of G.
		We show that the topology of the total space of JP is not T_1 by constructing a sequence of non-classical spreads in F^2 that converges to the classical spread in F^2, where F^2 \in {\CC,\HH,\OO}.
	www.mathematik.uni-tuebingen.de /ab/Geometrie.alt/Abstracts.html (1815 words)

	excision.txt
		He* re, spaces of holomorphic maps are given the compact-open* topology; it is for technical re* *asons that we turn them into simplicial sets by applying the singular functor s.
		Using the* * compact- open topology allows us to work at a relatively simple technical level: we neit* *her have to localize nor stabilize to get something interesting.
		As before, we equip each set of holomorphic m* aps with the compact-open* topology, apply the singular functor, and obtain an embed* ding of A into the model category s ShvA of simplicial sheaves on A, taking a quasi-pro *jective manifold X to the simplicial sheaf sO(.; X).
	www.math.purdue.edu /research/atopology/Larusson/excision.txt (2966 words)

	hypercover.txt
		Let U be an open cover of a space X with the property that every finite intersection Ua0...anis covered by other elements of U. Form the diagram consisting of all the Ua's and all the inclusions between them.
		An open cover U = {Ua} of a space X is called complete if for all finite sets oe of indices, the intersection Uoeis covered by elements of U.* * It is called a ~Cech cover if every Uoeis again an element of the cover.
		Proof of Theorem 2.1.Given any open set V in X, the space ß-1(V) is homeomor- phic to the space C~(U0), where U0 is the open* cover {Ua \ V } of the space * *V.
	www.math.purdue.edu /research/atopology/Dugger-Isaksen/hypercover.txt (7317 words)

	Haruto Ohta
		X is the set X equipped with the coarsest topology for which every Z -valued functions which is continuous continuous on every compact set is continuous.
		A Z -compact space is a space which is homeomorphic to a closed subspace of a power of Z.
		Topology Atlas Conference Abstracts Document # caai-82.htm
	www.utm.edu /staff/jschomme/topology/c/a/a/i/82.htm (235 words)

	York Topology Seminars 96/97
		Lucia Junqueira- The Topology of Elementary Submodels I
		A small and informal workshop on Applying Elementary Submodels to General Topology will be held this week at York.
		Lucia Junqueira- The Topology of Elementary Submodels II N638 Ross 10:45 AM
	www.math.yorku.ca /Seminars/Topology/a96.html (266 words)

	Atlas: Topological annihilators in compact abelian groups by Gabor Lukacs
		The Pontryagin dual of an abelian topological group G is the group [^G]=\mathscrH(G, T) of all continuous characters of G equipped with the compact-open topology (T is the circle group).
		Sequences and filters of characters characterizing subgroups of compact abelian groups.
		on the discrete group A=[^K] (namely, the topology of pointwise convergence on H).
	atlas-conferences.com /cgi-bin/abstract/capa-79 (334 words)

	Algebraic Topology: Homotopy
		[E.g., give C(X,Y) the compact-open topology, and assume that X is a k-space, i.e., has the topology defined by the injection maps from its compact subspaces.
		Let L(X;x) be the space of loops in X with base point x (i.e., maps from I to X sending 0 and 1 to x), provided with the compact-open topology.
		For suitably nice (say, locally compact) X we have pi_2(X;x) = P(L(X;x);e) when e is the constant map that sends I to x, and more generally pi_n(X;x) = P(L_(n-1)(X;x);e) where L_n(X;x) = L(L_(n-1)(X;x);e) for a suitable constant e = e_n.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	M. Montserrat Bruguera
		If G is a LCA group, the continuous convergence structure in \Gamma G is precisely the convergence given by the compact open topology [3], thus, the "convergence dual" and the ordinary dual are identical.
		If addition in \Gamma G is defined pointwise, then \Gamma G endowed with the compact open topology is a topological abelian group, which will be called G
		Examples of reflexive groups which are not locally compact are known from the late forties.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/10.htm (924 words)

	cpt_open
		Keywords: Compact-open topology Felix Dilke wrote: > > Can anyone tell me what ways there are to > topologize the set Hom(X, Y) of all continuous > maps X->Y, for two topological spaces X & Y? > I'm mainly interested in the case where X, Y are > compact Hausdorff.
		If K is a compact subset of X and U an open subset of Y let A(K,U) be the set of continuous f:X --> Y with f(K) contained in U. Then the A(K,U) form a subbasis for a topology on the function space -- the compact-open topology.
		One of the most popular is the compact-open topology.
	www.math.niu.edu /~rusin/known-math/99/cpt_open (243 words)

	Atlas: Topologizing homeomorphism groups of rim-compact spaces by Anna Di Concilio
		When X is further finite union of disjoint connected subspaces, then the minimum is closely linked to its Freudenthal compactification and it is the closed-open topology determined from all closed sets with compact boundaries.
		, the existence of the minimum among all admissible topological group structures on H(X) which can be described simply as a set-open topology, further agreeing with the compact-open topology when X is locally connected.
		In the rational case again the minimum is linked to the Freudenthal compactification of Q and it is just the closed-open topology.
	atlas-conferences.com /cgi-bin/abstract/cait-41 (332 words)

	Re: unitary equivalence for C*-algebras
		Moreover, if K is sigma-compact, then this compact-open topology is metrizable (so C(K) is an F-space --- if you close it you get a Frechet space).
		And, if I can embed any algebra in C(K) with the compact-open topology, I can extend the functional calculus to any abelian algebra simply by using C(K) to define the effect of applying a continous function to any element of the algebra.
		In article , wrote: >In article , >Miguel Carrion wrote: > >>Every abelian C* algebra is isomorphic to the algebra C(K) of complex >>continuous functions on the compact space K, and the norm corresponds >>to the topology of uniform convergence.
	www.lns.cornell.edu /spr/2003-07/msg0052818.html (978 words)

	The page cannot be found
		Open IIS Help, which is accessible in IIS Manager (inetmgr), and search for topics titled Web Site Setup, Common Administrative Tasks, and About Custom Error Messages.
		Go to Microsoft Product Support Services and perform a title search for the words HTTP and 404.
	www.absoluteastronomy.com /encyclopedia/c/co/compact-open_topology.htm (121 words)

	DMS.MPS.a9704849.txt
		Title : Problems in Set-Theoretic Topology Abstract : The principal investigator will continue his research on several open problems in set-theoretic topology.
		Second, two problems concerning closed mappings will be investigated, the first being the well-known open question about the preservation of the metalindelof property under closed or perfect maps, and the other a question about the existence of irreducible restrictions of closed mappings.
		Objects of study in topology include familiar spaces such as the plane and ordinary three-dimensional Euclidean space, and their subsets, as well as much more abstract spaces.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9704849.txt (319 words)

	HJM, Vol. 31, No. 1, 2005
		The external ones involve topological properties of the space of real-valued continuous functions over the manifold, endowed either with the topology of pointwise convergence or with the compact-open topology.
		We establish that, for any locally compact subgroup H of a topological group G, the natural quotient mapping of G onto the quotient space G/H is locally perfect, that is, the restriction of it to the closure of some open set is an open mapping with compact fibers.
		If T is an integral commutative extension of a ring R such that R is an open ring, R[a, b] is a going-down ring for each a, b in T and T is semiquasilocal, then each ring contained between R and T is an open ring.
	www.math.uh.edu /~hjm/Vol31-1.html (1971 words)

	Preliminary Examination in Topology
		Compactness and related topics like local compactness, paracompactness, and the Tychonoff theorem.
		Complete metric spaces, including Cauchy sequences and compactness in metric spaces.
		The fundamental group, covering spaces, and the relations between them.
	www.ms.uky.edu /~ochanine/TopPrelim_files/Topics.htm (106 words)

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