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	PlanetMath: topology induced by uniform structure
		be the entourage whose existence is granted by the definition of open set.
		"topology induced by uniform structure" is owned by n3o.
		This is version 2 of topology induced by uniform structure, born on 2002-06-10, modified 2002-06-11.
	planetmath.org /encyclopedia/TopologyInducedByAUniformStructure.html (83 words)

	Topology glossary (Site not responding. Last check: 2007-09-08)
		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.
		If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A.
		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.
	www.factsite.co.uk /en/wikipedia/t/to/topology_glossary.html (3661 words)

	Read about Topology glossary at WorldVillage Encyclopedia. Research Topology glossary and learn about Topology glossary ... (Site not responding. Last check: 2007-09-08)
		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.
		The topology T is the smallest topology on X containing B and is said to be generated by B.
		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.
	encyclopedia.worldvillage.com /s/b/Topology_glossary (3927 words)

	Uniform space (Site not responding. Last check: 2007-09-08)
		In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence.
		Uniform spaces may be defined alternatively and equivalently using systems of pseudo-metrics, an approach which is often useful in functional analysis.
		Every uniform space is a completely regular topological space, and conversely, every completely regular space can be turned into a uniform space (often in many ways) so that the induced topology coincides with the given one.
	www.factsite.co.uk /en/wikipedia/u/un/uniform_space.html (696 words)

	Business Software Review : Article 'Alexandrov topology' (Site not responding. Last check: 2007-09-08)
		In general topology the open sets of a topological space satisfy by definition the conditions: The union of arbitrarily many open sets is open.
		If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A. This construction is dual to the construction of the quotient topology.
		Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.
	www.business-software-review.org /DisplayArticle451854.html (6849 words)

	Read about Trivial topology at WorldVillage Encyclopedia. Research Trivial topology and learn about Trivial topology ... (Site not responding. Last check: 2007-09-08)
		The trivial topology is the topology with the least possible number of open sets, since the definition of a topology requires these two sets to be open.
		Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.
		In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.
	encyclopedia.worldvillage.com /s/b/Indiscrete_space (435 words)

	Neighbourhood (mathematics) - Wikpedia (Site not responding. Last check: 2007-09-08)
		In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
		Intuitively speaking, a neighbourhood of a point is a set containing the point where you can "wiggle" or "move" the point a bit without leaving the set.
		In a uniform space S:=(X, δ) V is called a uniform neighbourhood of P if P is not close to X \ V, that is there exists no entourage containing P and X \ V.
	bostoncoop.net /~tpryor/wiki/index.php?title=Neighbourhood_(topology) (331 words)

	Discrete space - Free Online Information (Site not responding. Last check: 2007-09-08)
		In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.
		The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
		For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups.
	www.sneaknews.com /econtents/di/Discrete_space.html (797 words)

	Discrete space (Site not responding. Last check: 2007-09-08)
		Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every point in Y has a neighborhood on which f is constant.
		In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the least possible number of open sets (just the empty set and the space itself).
		Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.
	www.worldhistory.com /wiki/D/Discrete-space.htm (1071 words)

	PlanetMath: uniform structure of a metric space
		, which induces the same topology as the metric.
		to be an entourage if and only if it contains a subset of the form
		This is version 2 of uniform structure of a metric space, born on 2002-06-13, modified 2002-06-15.
	planetmath.org /encyclopedia/UniformStructureOfAMetricSpace.html (82 words)

	D. Doitchinov (Site not responding. Last check: 2007-09-08)
		Topology Atlas Conference Abstracts Document # caaj-142.htm
		The purpose of this paper is to give a construction of a special Cauchy completion of a uniform frame.
		As for the notion of a uniform frame itself, we use here, in a slightly modified version, its "entourage" definition proposed in [3] which instead of coverings, earlier used for the same purpose, uses mappings of a special kind.
	www.utm.edu /staff/jschomme/topology/c/a/a/j/142.htm (172 words)

	Abstract (Site not responding. Last check: 2007-09-08)
		Abstract: In pointfree topology, the notion of uniformity in the form of a system of covers was introduced by J. Isbell in [Atomless parts of spaces, Math.
		Another equivalent notion of locale uniformity was given by P. Fletcher and W. Hunsaker in [Entourage uniformities for frames, Monatsh.
		The purpose of this paper is to formulate and investigate an alternative definition of entourage uniformity which is more likely to the Weil pointed entourage uniformity, since it is expressed in terms of products of locales.
	www.mat.uc.pt /~picado/publicat/9405.html (170 words)

	Uniform space explained (Site not responding. Last check: 2007-09-08)
		Category:Topology In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence.
		The elements of Φ are called entourages (french:neighborhoods).
		The essential difference between a topological space and a uniform space is that in a uniform space, you can formalize the idea that "''x''
	www.wordspider.net /un/uniform-space.html (563 words)

	Citations: Completeness of quasi-uniform and syntopological spaces - Smyth, College (ResearchIndex) (Site not responding. Last check: 2007-09-08)
		The appropriate definition for continuity spaces is as follows: Definition 7 A topological V continuity space (tV cs) is a triple (X; d; where (X; d) is a V continuity space and is a....
		A Cauchy net on a quasi uniform space (X; U) is a net (x i) i2I with the property that for every entourage U 2 U, there is some index i 2 I such that for all indices j, k with i j k the relation x j U x k holds.
		The appropriate definition for continuity spaces is as follows: Definition 4 A topological V continuity space (tV cs) is a triple (X; d; where (X; d) is a V continuity space and is a....
	citeseer.ist.psu.edu /context/186256/0 (2374 words)

	Uniform space (Site not responding. Last check: 2007-09-08)
		Every metric space (M, d) can be considered as a uniform space by defining a subset V of M × M to be an entourage if and only if there exists an ε > 0 such that for all x, y in M with d(x, y) < ε we have (x, y) in V.
		This uniform structure on M generates the usual topology on M.
		Every topological group (G,) becomes a uniform space if we define a subset V of G × G to be an entourage* if and only if the set {x*y
	www.factbase.info /un/uniform-space.html (620 words)

	Hans-Peter A. Kuenzi, Nonsymmetric distance structures
		Many classical counterexamples in topology like the Sorgenfrey plane, the Niemytzki plane, the Michael line or the Pixley-Roy space over the reals are (obviously) quasi-metrizable.
		Later they were dealt with by Niemytzki when he explored the interplay of the various assumptions in the usual axiomatization of a metric space.
		(A paratopological group is a group equipped with a topology such that the group operation is continuous.) Recently further applications of methods from the theory of quasi-uniform spaces to problems in approximation theory and functional analysis were discovered.
	at.yorku.ca /t/a/i/c/37.htm (1164 words)

	physics - Trivial topology
		In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.
		The trivial topology belongs to a pseudometric space in which the distance between any two points is zero, and to a uniform space in which the whole cartesian product X × X is the only entourage.
		If F : Top → Set is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and G : Set → Top is the functor that puts the trivial topology on a given set, then G is right adjoint to F.
	www.physicsdaily.com /physics/Indiscrete_space (449 words)

	entourage - OneLook Dictionary Search
		entourage : Compact Oxford English Dictionary [home, info]
		Entourage : Online Plain Text English Dictionary [home, info]
		Words similar to entourage: cortege, retinue, suite, more...
	www.onelook.com /?w=entourage (171 words)

	OSXFAQ :: View topic - Problem with Entourage and Airport Extreme base station
		I'm having a problem getting Entourage to connect to my Pop3 mail server through my wireless airport extreme.
		If I plug my laptop directly into the airport with an ethernet cable, then I can connect just fine, but if I try to connect wirelessly it gives me an error about not being able to establish a connection.
		It seems to me it should be very simple to get this working, but for some reason it just cannot connect through my wireless connection.
	forums.osxfaq.com /viewtopic.php?t=10403 (837 words)

	Atlas: Localic entourage uniformities revisited by Jorge Picado (Site not responding. Last check: 2007-09-08)
		Atlas: Localic entourage uniformities revisited by Jorge Picado
		Classically the description of uniform structures via entourages is based on the calculus of relations on sets, that is, the 2-category of relations on sets.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-47.
	atlas-conferences.com /c/a/g/x/47.htm (205 words)

	Topics: Uniformities
		Idea: A richer structure than a topology, due to A Weil (1937, entourage version) and Tukey (covering version).
		Examples: p-adic structure; the additive uniformity on R defined by V
		And other structure: A topological space is uniformizable iff it is completely regular (also, a gage space); Viceversa, the topology defined by a uniformity is always completely regular; A uniformity defines a proximity by A
	www.phy.olemiss.edu /~luca/Topics/u/uniformity.html (173 words)

	kbAlertz: Entourage X for Mac Knowledge Base Articles
		If you login (using the form above) or create an account your searches will be saved and displayed here.
		Want to see the top articles for Entourage X for Mac?
		This article describes the recommended installation procedure for Exchange 2000 Service Pack 1 (SP1) when your Exchange 2000 computers are configured in a front-end/back-end topology.
	www.kbalertz.com /technology_273_393.aspx (804 words)

	Atlas: International Conference on Applicable General Topology - List of Speakers
		Guillaume Brümmer Some problems in categorical and asymmetric topology
		A review of new results and some new developments.
		Giovanni Sambin The basic picture, meet and definitions in predicative topology
	atlas-conferences.com /c/a/g/x/01.htm (432 words)

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