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Topic: Paracompact Hausdorff space

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	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover.
		On paracompact Hausdorff spaces, the cohomology of a sheaf is equal to its Äech cohomology.
		For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=paracompact (1400 words)

	Topology (Site not responding. Last check: 2007-10-15)
		In 1914, Felix Hausdorff, generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space.
		The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
		A metric space is Hausdorff, also normal and paracompact.
	hallencyclopedia.com /Topology (1542 words)

	Normal space
		X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint.
		Sierpinski space[?] is an example of a normal space that isn't regular.
		A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence[?].
	www.ebroadcast.com.au /lookup/encyclopedia/no/Normal_Hausdorff_space.html (908 words)

	Springer Online Reference Works
		The significance of paracompactness is shown by the clear generality of this concept and by a number of remarkable properties of paracompacta.
		Paracompactness is not inherited by arbitrary subspaces (in which it differs from metrizability), otherwise, for example, all Tikhonov spaces, as subspaces of Hausdorff compacta, would be paracompact.
		On the other hand, in the class of Hausdorff spaces, the inverse image of a paracompactum under a perfect mapping is a paracompactum, and the image of a paracompactum under a continuous closed mapping is a paracompactum.
	eom.springer.de /P/p071300.htm (1097 words)

	Topology - Wikipedia
		Any metric space turns into a topological space if we define a set to be open if it is a (possibly infinite) union of open balls.
		In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
		On a paracompact Hausdorff space every open covering admits a partition of unity subordinate to the cover.
	nostalgia.wikipedia.org /wiki/Topology (941 words)

	PlanetMath: paracompact topological space
		Any metric or metrizable space is paracompact (A. Stone).
		Cross-references: pseudometric space, compact, regular, Hausdorff space, cover, partition of unity, metrizable space, metric, properties, open subsets, open refinement, locally finite, open cover, topological space
		This is version 5 of paracompact topological space, born on 2002-01-22, modified 2007-06-24.
	planetmath.org /encyclopedia/Paracompact.html (82 words)

	Paracompact space (Site not responding. Last check: 2007-10-15)
		Note the similarity between the definitions of compact and paracompact: for paracompact we replace by "open refinement" and "finite" by "locally Both of these changes are significant: if take the above definition of paracompact and "open refinement" back to "subcover" or "locally back to "finite" we end up with compact spaces in both cases.
		For instance the integral of differential forms on paracompact manifolds is first defined locally (where the looks like Euclidean space and the integral is well known) this definition is then extended to the space via a partition of unity.
		Any space is fully normal must be paracompact and space that is paracompact must be metacompact.
	www.freeglossary.com /Paracompact (1211 words)

	Reference.com/Encyclopedia/Paracompact space
		As you might guess from the generality of most of the examples above, it is actually harder to think of spaces that are not paracompact than to think of spaces that are.
		This means the following: if X is a paracompact Hausdorff space with a given open cover, then there exists a collection of continuous functions on X with values in the unit interval [0, 1] such that:
		This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).
	www.reference.com /browse/wiki/Paracompact (1470 words)

	Paracompact Article, Paracompact Information (Site not responding. Last check: 2007-10-15)
		The most important feature of paracompact Hausdorff spaces is thatthey are normal and admit partitions of unity relative to any open cover.
		For instance,the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is thenextended to the whole space via a partition of unity.
		For example,manifolds are usually defined to be paracompact, thus allowing integration of differential forms to be defined as in the previoussection, while excluding the long line, which is useless in almost every application.
	www.anoca.org /space/cover/paracompact.html (903 words)

	Metric space - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-15)
		In metric spaces, one can talk about limits of sequences; a metric space in which every Cauchy sequence has a limit is said to be complete.
		Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal).
		An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem).
	xahlee.org /_p/wiki/Metric_spaces.html (1390 words)

	Learn more about Topology in the online encyclopedia. (Site not responding. Last check: 2007-10-15)
		In mathematics, topology is a branch concerned with the study of topological spaces.
		In 1914, Hausdorff coined the term "topological space" and gave definition to what is now called Hausdorff space.
		The current concept of topological space was described by Kuratowski in 1922.
	www.onlineencyclopedia.org /t/to/topology_1.html (1178 words)

	Topology (Site not responding. Last check: 2007-10-15)
		In mathematics, topology is a branch concerned with thestudy of topological spaces.
		In 1914, Felix Hausdorf, generalizing the notion of metric space, coined theterm "topological space" and gave the definition for what is now called Hausdorff space.
		The traditional joke is that the topologist can't tell the coffeecup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of acoffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
	www.therfcc.org /topology-89.html (1256 words)

	A.P. Kombarov
		Recall that a space is said to be pseudonormal if every countable closed subset has arbitrarily small closed neighborhoods.
		A space is called to have property D [4] if every countable closed discrete set has arbitrarily small closed neighborhoods.
		The exponential space \exp(X) is the set of all non-empty closed subsets of X with Vietoris (finite) topology.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/55.htm (697 words)

	Topology
		Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing).
		The spaces studied in topology are called topological spaces.
		Georg Cantor, the inventor of set theory along with Gottlob Frege, had begun to study the theory of point sets in Euclidean space, in the later part of the 19th century.
	www.cooldictionary.com /words/Topology.wikipedia (1581 words)

	iqexpand.com (Site not responding. Last check: 2007-10-15)
		Look for Paracompact hausdorff space in the Commons, our repository for free images, music, sound, and video.
		fully normal space that is also T 1 ; see Separation axioms) is the same thing as a paracompact Hausdorff space.
		Paracompact Hausdorff space Paracompact space Paracompactness Paraconglomerates Paraconglomerates Paraconic Paraconine Paraconsistent analysis Paraconsistent logic Paraconsistent logics Paraconsistent...
	paracompact_hausdorff_space.iqexpand.com (392 words)

	topoeogy information,topology (Site not responding. Last check: 2007-10-15)
		In mathematics, topoeogy is a branch concerned with thestudy of topological spaces.
		In 1914, Felix Hausdorff, generalizing the notion of metric space, coined theterm "topological space" and gave the definition for what is now called Hausdorff space.
		Inthat case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes oftopoeogy.
	www.pin-outs.com /topoeogy.html (1312 words)

	Fibrewise General Topology: A Brief Outlook by David Buhagiar
		The carried out research showed a strong analogy in the behaviour of spaces and maps and it was possible to extend the main notions and results of spaces to that of maps.
		One can also add that the definitions of paracompact maps, metacompact maps, subparacompact maps and submetacompact maps strengthened the result that paracompactness, metacompactness, subparacompactness and submetacompactness are all inverse invariant of perfect maps.
		Two classes of spaces were considered in [7] as the collection C, the class of all metrizable spaces and the class of all linearly ordered topological spaces (i.e., LOTS).
	at.yorku.ca /t/a/i/c/34.htm (1505 words)

	Metrization theorems : Metrisable space
		A metrizable space is a topological space which is homeomorphic to a metric space.
		For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first countable.
		For example, a compact Hausdorff space is metrizable if and only if it is second-countable.
	www.fastload.org /me/Metrisable_space.html (326 words)

	Web page experiment
		research is focused on spaces in which a
		given space is dense (extensions), and on the related notion of
		she looked at tend to be closely related to Stone spaces of Boolean algebras.
	www.math.ku.edu /~roitman/seminar.html (259 words)

	toplogy information,topology (Site not responding. Last check: 2007-10-15)
		In mathematics, toplogy is a branch concerned with thestudy of topological spaces.
		General toplogy, or point-set toplogy, defines and studiessome useful properties of spaces and maps, such as connectedness, compactness and continuity.
		Inthat case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes oftoplogy.
	www.pin-outs.com /toplogy.html (1312 words)

	Paracompact space
		A paracompact space is a topological space in which every open cover admits an open locally finite refinement.
		In symbols, the cover V is a refinement of the cover U iff, for any
		Every metric space (or metrisable space) is paracompact.
	www.gamesinathens.com /olympics/p/pa/paracompact_space.shtml (791 words)

	PlanetMath: manifold
		A manifold is a space that is locally like
		Standard illustrations of these pathologies are given by the long line (lack of paracompactness) and the forked line (points cannot be separated).
		See Also: notes on the classical definition of a manifold, locally Euclidean, 3-manifold, surface, topological manifold, Lagrange multipliers on manifolds
	planetmath.org /encyclopedia/Manifold.html (415 words)

	Citebase - On strict category weight and Arnold conjecture (Site not responding. Last check: 2007-10-15)
		Let X be a Hausdorff paracompact space, and let u H q (X, G) be an arbitrary element.
		A flow on a topological space X is a family = {t }, t R where each t:X X is a self-homeomorphism and s t = s+t for every s, t R (notice that this implies 0 = 1X).
		Let be a continuous gradient-like flow on a compact metric space X, let Y be a Hausdorff space which can be covered by open and contractible in Y subspaces, and let f:X Y be a map.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/9806115 (1742 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		The first is that the space, which we should call \begin_inset Formula \(M \) \end_inset in general, has to be assumed to be \shape italic Hausdorff.
		Well, it may be possible that the same topological space can have \shape italic two or more \shape default distinct smooth structures.
		For years it was known that the topological space \begin_inset Formula \(\R ^{n} \) \end_inset had only \shape italic one \shape default structure, for \begin_inset Formula \(n\neq 4 \) \end_inset.
	www.lehigh.edu /dlj0/yesterday/Desktop/dlj0/courses/423f96-lect2.lyx (1632 words)

	Fixed points and selections of set valued maps on spaces with convexity (Site not responding. Last check: 2007-10-15)
		Let X be a paracompact Hausdorff topological space, and let Y be a Banach space.
		Let X be a paracompact Hausdorff topological space, and let Z be any topological vector space.
		We are able to construct a convexity structure for a wide class of topological spaces, which makes it possible to prove a generalization of the following purely topological fixed point theorem.
	users.marshall.edu /~saveliev/Research/FPT/FPT.htm (371 words)

	Topological Manifold (Site not responding. Last check: 2007-10-15)
		Hausdorff Space) and countability (i.e., it is a
		The first nonsmooth topological manifold occurs in 4-D. Nonparacompact manifolds are of little use in mathematics, but non-Hausdorff manifolds do occasionally arise in research (Hawking and Ellis 1975).
		For manifolds, Hausdorff and second countable are equivalent to Hausdorff and paracompact, and both are equivalent to the manifold being embeddable in some large-dimensional Euclidean space.
	mathserver.sdu.edu.cn /mathency/math/t/t179.htm (84 words)

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