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Topology glossary

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	Topology glossary - Wikipedia, the free encyclopedia
		The topology T is the smallest topology on X containing B and is said to be generated by B.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
		The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
	en.wikipedia.org /wiki/Topology_glossary (4544 words)

	Topological space
		The Zariski topology is a purely algebraically defined topology on the spectrum of a ring or an algebraic variety.
		Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.
		A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space.
	www.starrepublic.org /encyclopedia/wikipedia/t/to/topological_space.html (2106 words)

	Read about Topology glossary at WorldVillage Encyclopedia. Research Topology glossary and learn about Topology glossary ... (Site not responding. Last check: 2007-10-09)
		topology for a brief history and description of the subject area.
		Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed.
		If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of all unions of finite intersections of elements of B.
	encyclopedia.worldvillage.com /s/b/Topology_glossary (3927 words)

	Topology glossary
		Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and on definitions that are fundamental to a broad range of areas.
		The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed pointss of this operator, i.e.
		A collection of open sets is a subbase (or subbasis) for a topology if every open set in the topology is a union of finite intersections of sets in the subbase.
	www.starrepublic.org /encyclopedia/wikipedia/t/to/topology_glossary.html (3661 words)

	Reference.com/Encyclopedia/Comparison of topologies
		All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
		The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
		In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
	www.reference.com /browse/wiki/Coarser_topology (499 words)

	PlanetMath: relatively prime integer topology
		is the topology determined by a basis consisting of the sets
		"relatively prime integer topology" is owned by mathcam.
		This is version 3 of relatively prime integer topology, born on 2004-10-07, modified 2004-10-23.
	planetmath.org /encyclopedia/PrimeIntegerTopology.html (127 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)

	Sports Fresh : Article 'Normal space' (Site not responding. Last check: 2007-10-09)
		In topology and related branches of mathematics, normal spaces, T 4 spaces, and T 5 spaces are particularly nice kinds of topological spaces.
		The collection of all open balls of M is a base for a topology on M ; this is the topology on M induced by d.
		If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B ; this topology consists of all unions of finite intersections of elements of B.
	www.sports-fresh.net /DisplayArticle66422.html (5471 words)

	EZGeography - Topological space
		Note that the requirement that the union of any collection of sets is a member of the topology is more stringent than simply requiring that all pairwise unions must be members, as the former includes unions of infinite collections of sets.
		A topology is completely determined if for every net in X the set of its accumulation points is specified.
		This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology.
	www.ezgeography.com /encyclopedia/Topological_space (2366 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Roughly speaking, a topology is a way of specifying the concept of "nearness"; an open set is "near" each of its points.
		gets a topology where a set is open if and only if its inverse image is open.
		It is almost universally true that all "large" algebraic objects carry a natural topology which is compatible with the algebraic operations.
	www.online-encyclopedia.info /encyclopedia/t/to/topological_space.html (1204 words)

	[No title]
		Turned the other way ("a Hausdorff topology cannot be properly contained in a compact topology") it says Hausdorff topologies have to have kind of a lot of open sets.
		Taken together ("no two compact, Hausdorff topologies are comparable") this makes it clear that compact+Hausdorff is quite a restrictive condition, and thus one from which many nice results can be expected to follow (indeed, Bourbaki _defines_ compact to include the Hausdorff axiom).
		Undoubtedly there are non-trivial topologies on spaces X for which this Ibokor compactification would turn out to be the extreme case (X, coarse).
	www.math.niu.edu /~rusin/known-math/95/bijection (909 words)

	Re: Fiber Bundles and "Nearness" (Site not responding. Last check: 2007-10-09)
		But you suggested a different (much coarser) topology on TS^1, namely the coarsest topology that makes the bundle projection TS^1 = S^1 x R --> S^1 (given by (p,r) -> p) continuous: two points are near each other if and only if their projections in S^1 are near each other.
		There is of course a nice and canonical topology on the set of fibres: the bundle projection defines a bijective map between B and the set of fibres, and you can use this map to "copy" the topology of B to the set of fibres.
		This topology can (in general) not be reconstructed from the topology of B and the topology of F. It is determined by the bundle charts (i.e.
	www.lns.cornell.edu /spr/2001-12/msg0037335.html (887 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		A rigorous description of a context in terms of a mathematically formulated context-independent fundamental theory is possible by the restriction of the domain of the basic theory and the introduction of a new coarser topology.
		Such a new topology is never given by first principles, but depends in a crucial way on the abstractions made by the cognitive apparatus or the pattern recognition devices used by the experimentalist.
		Most intertheoretical relations are mathematically describable as singular asymptotic expansions which do not converge in the topology of the primary theory, or by choosing one of the infinitely many possible, physically inequivalent representations of the primary theory (Gelfand–Naimark–Segal-construction of algebraic quantum mechanics).
	www.cs.odu.edu /~melmilig/maktbaelectroneya/files/00000953.dc (252 words)

	Articles - Topological space (Site not responding. Last check: 2007-10-09)
		However, this is not in fact necessary, as the empty set is the union of the empty collection and must therefore be in T by rule 2.
		A variety of topologies can be placed on a set to form a topological space.
		: Î± in A} is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X which contain every member of F.
	www.beatlesa.com /articles/Topological_space (1561 words)

	Physics Help and Math Help - Physics Forums - View Single Post - Anoter question on vector spaces an rationnal numbers (Site not responding. Last check: 2007-10-09)
		Therefore, I may understand better the limit, with the adequate topology, of this sequence.
		For the basic set {0,1} and the sequence {0,1}_{n}, it is more complicated because, for any n, I have no bijection between N and {0,1}_{n} and it is more difficult to see the evolution from a finite (countable) set to an infinite non countable set.
		The ideal should be to find a sequence of infinite countable sets that converge (with the adequate topology) to a non countable set equivalent to the set of all Q Cauchy sequences or something like that (i.e.
	www.physicsforums.com /showpost.php?p=408538&postcount=14 (647 words)

	PlanetMath: topological space
		Cross-references: metric topology, product topology, subspace topology, indiscrete topology, power set, discrete topology, coarser, finer, complement, subsets
		it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws.
		i can't seem to find a definition for the topology induced by a metric space either, which is something you probably want to add.
	planetmath.org /encyclopedia/Topology.html (168 words)

	Citations: Continuity Spaces: Reconciling Domains and Metric Spaces - Flagg, Kopperman (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Proposition 6.3 For every subset V of a gum X, cl A (V) R A ffi ae A (V) Proof: It follows from the characterization (11) of R A ffi ae A that it is sufficient to prove cl A (V)....
		Proposition 6.3 For every subset V of a gms X, cl A (V) R A ffi ae A (V) 17 Proof: It follows from the characterization (10) of R A ffi ae A that it is sufficient to prove cl A....
		The Yoneda lemma and the generalized Alexandroff topology are discussed in Section 3, while the generalized Scott topology is presented in Section 5.
	citeseer.ist.psu.edu /context/227488/0 (2987 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		As wisely stated by Scarlatos, the placement of points in a regular tessellation is independent of the surface topology.
		When using regular grids, a simple solution is to force the higher resolution grid to meet the coarser one [Hughes 93].
		One drawback of this is that it alters the grid.
	gasa.dcea.fct.unl.pt /gasa/gasa98/gasa98/papers/mux/terr.htm (5233 words)

	Topology glossary (Site not responding. Last check: 2007-10-09)
		If is a collection of spaces and X is the (set-theoretic) disjoint union of, then the coproduct topology (or disjoint union topology, topological sum of the X
		More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if is open in the subspace topology on S.
		If is a collection of spaces and X is the (set-theoretic) product of, then the product topology on X is the coarsest topology for which all the projection maps are continuous.
	www.worldhistory.com /wiki/T/Topology-glossary.htm (4505 words)

	[No title]
		S are just diagrams of simplicial sets, and the option of choosing the chaotic topology (or rather, no topology at all) produces one of the standard diagram-theoretic homotopy theories.
		All other topologies on C are finer than the chaotic topology, and all produce different homotopy theories.
		Another observation is that, while it is true that different topologies dete* *r- mine different homotopy theories for simplicial presheaves on a fixed category C, the sheaves and simplicial sheaves on C are somehow beside the point, except that they give the means of specifying weak equivalences.
	hopf.math.purdue.edu /Jardine/gen-shea.txt (8175 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		>> >>With this topology, N* (the set of positive integers) >>is Hausdorff and connected, the sets E(a,b) are >>totally discontinuous (I'm not sure it's the right >>English term.) >> >>My friend conjectures that with this topology, the >>set of all prime numbers is connected.
		There are two variants of this topology, a finer one where (a, b) are not required to be relatively prime, and a coarser one where b must be prime.
		(The coarser topology mentioned is both connected and locally connected).
	www.math.niu.edu /~rusin/known-math/96/prime.top (469 words)

	Topological cyclic homology of schemes (Site not responding. Last check: 2007-10-09)
		We use Thomasons's hypercohomology construction to extend the definition of topological cyclic homology to schemes.
		We justify this definition by showing that it agrees with the previous definition for affine schemes, and show that it does not depend on the topology coarser than the etale topology.
		For smooth schemes over perfect fields of characteristic p we identify the topological cyclic homology sheaf for the Zariski and etale topology; in the etale topology it agrees with the p-completed K-theory sheaf.
	www-math.mit.edu /~larsh/papers/008 (117 words)

	[No title]
		For this to be a topology, we need the directedness condition, that given F* * aM and F bM, there exists c such that F cM F aM \ F bM.
		We need a somewhat coarser topology that has better properties and a better cha* *nce of making d in (4.5) a homeomorphism.
		Indeed, it is the coarsest natural topology that makes E(X) discrete for all finite X. Of course, it coincides with the skeleton topology* when X has finite type.
	hopf.math.purdue.edu /Boardman/stabop.txt (21024 words)

	AMCA: Coarser connected topologies of Hausdorff spaces by Judith Roitman (Site not responding. Last check: 2007-10-09)
		Many extensions of \omega have coarser connected topologies, e.g., thin-tall scattered spaces.
		There is X with a coarser connected topology but its semi-regularization X(s) does not have a coarser connected topology.
		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/g/w/12.htm (175 words)

	Citations: Quasi-uniformities: Reconciling domains with metric spaces - Smyth (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Both topologies are defined in two ways: by giving the open sets and by a closure operator.
		This and motivation from domain theory leads to the investigation of quasi uniform spaces with an additional topology which may, but need not coincide with the induced one.
		....quasi metric on the space for example, it does not in the poset case, where the latter is the Alexandroff topology.
	citeseer.ist.psu.edu /context/186255/0 (3594 words)

	mp_arc 04-14 (Site not responding. Last check: 2007-10-09)
		We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t.
		which it is amenable, viz the relative weak topology of C(X,M(n)).
		This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=04-14 (82 words)

	NTU Info Centre: Topology glossary (Site not responding. Last check: 2007-10-09)
		A point x is an isolated point if the singleton {xis open.
		More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {xis open in the subspace topology on S.
		then the product topology on X is the coarsest topology for which all the projection maps are continuous.
	www.nowtryus.com /article:Partition_of_unity (4072 words)

	SemProba : vol. 16 [supplément] (Site not responding. Last check: 2007-10-09)
		Let E be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps C_k (the continuous functions with compact support) into continuous bounded functions.
		Let F be a compact space containing E as a dense subset, but inducing possibly a coarser topology.
		It is assumed that all potentials Uf with $f\in C_k$ extend to continuous functions on F, and that points of F are separated by continuous functions on F whose restriction to E is supermedian.
	www-irma.u-strasbg.fr /irma/semproba/SP1/sp-vol02.html (826 words)

	iqexpand.com (Site not responding. Last check: 2007-10-09)
		When every set in a topology T 1 is also in a topology T 2, we say that T 2 is finer than T 1, and T 1 is coarser than T 2.
		Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology.
		2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
	topological_space.iqexpand.com (2634 words)

	Amazon.com: An Invitation to Algebraic Geometry: Books (Site not responding. Last check: 2007-10-09)
		You do need a bit of topology and analysis to follow it.
		2 is a bit fast for someone unfamiliar with the subject), and a fair amount of topology.
		When I first got it, I read the first several pages and found them readable, but when I read more (on the car-ride home) I was confronted with the fact that the Zariski topology is coarser than the standard topology on C^n, and is not even Hausdorff.
	www.amazon.com /exec/obidos/tg/detail/-/0387989803?v=glance (1280 words)

	5. Singularities and standard bases
		However, we may also consider the local ring
		in an arbitrary small neighbourhood of 0 in the Euclidean topology while
		in an arbitrary small neighbourhood of 0 in the (much coarser) Zariski topology.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/29/paper_html/node8.html (673 words)

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