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Topic: Paracompact space

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	Paracompact space
		Note the similarity between the definitions of compact and paracompact: for paracompact we replace "subcover" by "open refinement" and "finite" by "locally finite".
		The most important feature of paracompact Hausdorff spaces is that they 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 then extended to the whole space via a partition of unity.
	www.ebroadcast.com.au /lookup/encyclopedia/pa/Paracompact_space.html (770 words)

	PlanetMath: paracompact topological space
		Any metric or metrizable space is paracompact (A. Stone).
		This is version 3 of paracompact topological space, born on 2002-01-22, modified 2003-07-15.
		Object id is 1540, canonical name is Paracompact.
	planetmath.org /encyclopedia/Paracompact.html (76 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		Therefore, the concept of a feathered space is an extension of both the concept of a locally compact space and the concept of a metric space.
		The pre-image of a feathered space under a perfect mapping is a feathered space (in the class of Tikhonov spaces).
		The image of a paracompact feathered space under a perfect mapping is a paracompact feathered space (Filippov's theorem); however, an example is known of a perfect mapping of a feathered space onto a non-feathered Tikhonov space.
	eom.springer.de /F/f038310.htm (633 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		Hausdorff spaces that have the latter property are said to be paracompact (cf.
		The existence in a regular space of a base that splits into a union of a countable family of locally finite open coverings is equivalent to the metrizability of this space.
		Open locally finite coverings of a normal space serve as a construction of a partition of unity on this space, subordinate to this covering.
	eom.springer.de /l/l060390.htm (541 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		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)

	Paracompact space - Wikipedia, the free encyclopedia
		Every metric space (hence, every metrisable space) is paracompact.
		This property is sometimes used to define paracompact spaces (at least in the Hausdorff case).
		Any space that is fully normal must be paracompact, any paracompact space must be metacompact; any metacompact space must be orthocompact.
	en.wikipedia.org /wiki/Paracompact (1301 words)

	Wikinfo \| Topological space
		The category of all topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics.
		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.
		A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
	www.wikinfo.org /wiki.php?title=Topological_space (2014 words)

	physics - Paracompact space (Site not responding. Last check: 2007-11-05)
		In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement.
		A regular space is paracompact if every open cover admits a locally finite refinement.
		For example, manifolds are usually defined to be paracompact, thus allowing integration of differential forms to be defined as in the previous section, while excluding the long line, which is useless in almost every application.
	www.physicsdaily.com /physics/Paracompact (868 words)

	directopedia : Directory : Science : Math : Topology
		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.
		In that case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes of topology.
		The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the doughnut 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.
	www.directopedia.org /directory/Science-Math/Topology.shtml (2012 words)

	Topology MAT 530
		The next were the formal definitions of metric and topological spaces, bases and subbases in topological spaces (i.e., a description of different ways to define topology on a set).
		A continuous invertible map from a compact space to a Hausdorff space is a homeomeorphism.
		The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B.
	www.math.sunysb.edu /~azinger/mat530/fall04/index.htm (2907 words)

	Inttroduction (Site not responding. Last check: 2007-11-05)
		A space is normal if every pair of disjoint closed sets can be separated, binormal if its product with the closed unit interval I is normal, and countably paracompact if every countable open cover has a locally finite open refinement.
		A Dowker space is a normal space that is not countably paracompact.
		An anti-Dowker space is a countably paracompact, (regular) space that is not normal.
	at.yorku.ca /b/a/a/b/09.l2h/index.htm (387 words)

	Amazon.com: "paracompact spaces": Key Phrase page (Site not responding. Last check: 2007-11-05)
		Recall that a space has point-countable type if every point is contained in a compact...
		Stone [1] showed that each metriz- able space is paracompact (a special case of this...
		PARACOMPACT SPACES 21 of bifurcation there is no neighbourhood in B whose pre-image in E' is a product.
	www.amazon.com /phrase/paracompact-spaces (437 words)

	Atlas: Spaces with property pp by Abdul Mohamad (Site not responding. Last check: 2007-11-05)
		The space X is said to have property pp if every open cover has a pp refinement.
		space then either of the words `closed' and `discrete' (but not both) may be removed from this definition.
		Under certain circumstances property pp is equivalent to paracompactness, in particular first countable pp spaces are paracompact.
	atlas-conferences.com /cgi-bin/abstract/capa-20 (178 words)

	[No title]
		Thus a space is paracompact iff each connected component is. Every locally compact and second countable Hausdorff space, in particular thus every connected second countable manifold, is paracompact; see for instance [2], section V.4.) An analytic manifold means a real-analytic paracompact manifold.
		The tangent space to the manifold M at the point, will be denoted by T,M, and T M is the tangent bundle to M. By a submanifold N of a manifold M we mean an immersed submanifold.
		Equivalently, because of the paracompactness assumption and the assumption of second countability that we shall make on the state spaces, we will be interested in determining when these submanifolds have only countably many components (in the submanifold topology).
	www.math.rutgers.edu /~sontag/actions-report.html (14799 words)

	[No title]
		Her space is large; it is a $P$-space ($G\sb \delta$ are open) and it is of size and weight $\aleph\sp \omega\sb \omega$.
		Instead of putting the subject to rest, Rudin's construction gave rise to a flurry of activity in search of a `small' Dowker space, that is, one whose important cardinal functions (density, character, size) are small.
		Separable, first-countable Dowker spaces of size continuum and $\omega\sb 1$ have been constructed under a variety of extra set-theoretic assumptions including the continuum hypothesis (CH), MA, the existence of a Suslin tree, and variations of $\lozenge$.
	www.math.niu.edu /~rusin/known-math/98/dowker (734 words)

	MathLinks Math Forums :: View topic - Product with every paracompact is paracompact => compact?
		Also, I suspect that there are spaces of different type whose product with every paracompact space is paracompact (though I do not see any example immediately).
		A Lindelof topological space is a topological space with the property that every open cover has a countable subcover.
		It is clear that any space that is a countable union of compact sets (such spaces are termed countable at infinity) satisfes the property that its product with every Lindelof space is Lindelof.
	www.mathlinks.ro /Forum/viewtopic.php?t=98117 (542 words)

	Proceedings of the American Mathematical Society
		A paracompact space is locally compact if and only if its product with every compact space is base-cover paracompact.
		A. Arhangelski, On the metrization of topological spaces, Bull.
		John O'Farrell, Construction of a Hurewicz metric space whose square is not a Hurewicz space, Fund.
	www.ams.org /proc/2004-132-10/S0002-9939-04-07457-X/home.html (436 words)

	Atlas: Some observations on quotients of topological groups with respect to locally compact subgroups. by Alexander ... (Site not responding. Last check: 2007-11-05)
		This result allows to prove that a number of topological properties are transfered from the quotient space G/H to the topological group G provided that H is locally compact.
		Suppose that G is a topological group, and H a locally compact subgroup of G such that the quotient space G/H is metrizable.
		Suppose that G is a topological group, and H a locally compact subgroup of G such that the quotient space G/H is (locally) paracompact.
	atlas-conferences.com /c/a/j/z/73.htm (384 words)

	Amazon.com: "paracompact space": Key Phrase page (Site not responding. Last check: 2007-11-05)
		Then the following statements are equivalent: a) X is a metrizable space, b) X is a paracompact space.
		The vector bundle structure is so rich that the set of isomorphism classes of k-dimensional vector bundles over a paracompact space B is in a natural bijective...
		Let X be a paracompact space and let be a (P2-ultrafilter on X. Assume does not converge.
	www.amazon.com /phrase/paracompact-space (521 words)

	Combinatorial Tiling Theory (Site not responding. Last check: 2007-11-05)
		Recall that a d-dimensional manifold is a topological space that is locally homeomorphic to E^d.
		Thurston (1980) introduced the concept of an orbifold as a Hausdorff, paracompact space O which is locally homeomorphic to the quotient space of R^d by a finite group action.
		In "Tiling Space By Platonic Solids I" (Delgado and Huson 1997), we show that there exist precisely 46, 58, and 914 equivariant types of tile-transitive tilings of by topological tetrahedra, octahedra, and cubes, falling in to 9, 3, and 11 topological families, respectively.
	www.math.uni-bielefeld.de /~huson/approach.html (3565 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Also, a paracompact space X is locally compact if and only if its product with every compact space is base-cover paracompact.
		Another property (base-family paracompactness) is used to characterize metrizability in the following way: A space X is metrizable if and only if its product with a converging sequence is base-family paracompact.
		In the past I have done research related to paracompactness properties, separation axioms, non-standard topologies on Minkowski space-time, and cardinal functions of topological spaces.
	www.ucs.louisiana.edu /~pgs2889/research (290 words)

	Topics: Types of Topological Spaces
		Def: The space (X, T) is 1st countable if for each pt p in X there is a countable collection of open sets s.t.
		Relationships: Every second countable space is first countable, Lindelöf, paracompact.
		Stone space: A "Boolean space", a totally disconnected compact Hausdorff space; Dual to Boolean algebras.
	www.phy.olemiss.edu /~luca/Topics/t/top_types.html (486 words)

	Paracompact and Metrizable Spaces (Site not responding. Last check: 2007-11-05)
		Next we define a locally finite subset family of a topological space and a paracompact topological space.
		An open sets family of a metric space we define next and it has been shown that the metric space with any open sets family is a topological space.
		Families of subsets, subspaces and mappings in topological spaces.
	www.cs.ualberta.ca /~piotr/Mizar/mirror/http/JFM/Vol3/pcomps_1.html (112 words)

	USC, Department of Mathematics: Seminars and Colloquia (Site not responding. Last check: 2007-11-05)
		A construction of a countably paracompact, first countable, collectionwise Hausdorff space that is not strongly collectionwise Hausdorff.
		The space is Tychonoff, and in Tychonoff spaces the latter property is equivalent to:
		Whenever f is a continuous function from a closed discrete subspace D to a Hilbert (or Banach) space, then f can be extended to the whole space.
	www.math.sc.edu /events/1999/events990927.html (328 words)

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