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Topic: Paracompactness

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	Paracompact space - Wikipedia, the free encyclopedia
		Note the similarity between the definitions of compact and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite".
		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.
		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.
	en.wikipedia.org /?title=Paracompact_space (1002 words)

	Paracompact space - Wikipedia, the free encyclopedia
		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.
		The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unity relative to any open cover.
	en.wikipedia.org /wiki/Paracompact_space (1002 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.
	www.planetmath.org /encyclopedia/Paracompactness.html (76 words)

	paracompactness (Site not responding. Last check: 2007-11-05)
		A paracompact space is a topological space in which every open cover admits an open locally finite refinement.
		Every locally compact second countable space is paracompact.
		Every metric space (or metrisable space) is paracompact.
	www.yourencyclopedia.net /Paracompactness.html (822 words)

	AMCA: Monotonizing countable paracompactness by Lylah Haynes (Site not responding. Last check: 2007-11-05)
		MCP is an interesting property since in many cases, the set-theoretic assumptions necessary for results concerning countable paracompactness may be abandoned with MCP.
		In this way MCP is analogous to monotone normality; set-theoretic assumptions may be dropped when replacing 'normal' with 'monotonically normal' in results concerning separation of closed discrete families in such spaces.
		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/m/c/86.htm (189 words)

	Manifold (Site not responding. Last check: 2007-11-05)
		It can be shown that a manifold is metrizable if and only if it is paracompact.
		Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an
		Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.
	www.sciencedaily.com /encyclopedia/manifold_1 (1402 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		So I'd have to say paracompactness is pretty important if you > want strong theorems which will, of course, only hold for "nice" spaces > (that's the kind of theorem I prefer, over weak-but-general theorems).
		So that's what paracompactness > does: it allows a passage from local to global.
		> > On the other hand, paracompactness is not of much help in algebraic topology, > except perhaps for preparing other conditions which in turn are more > suited for algebraic topology.
	www.math.niu.edu /~rusin/known-math/95/paracompact (245 words)

	Profiles of Women in Mathematics: Mary Ellen Rudin (Site not responding. Last check: 2007-11-05)
		Rudin's primary research area is set-theoretic topology, and she is particularly well known for her ability to construct counterexamples.
		Her Noether Lecture discussed several set-theoretic questions related to paracompactness.
		It is undecidable in Zermel-Frankel set theory whether there is a perfectly normal nonmetrizable manifold, and the question of whether every normal Moore space is metrizable has a more complex, unsatisfactory answer.
	www.awm-math.org /noetherbrochure/Rudin84.html (381 words)

	Citations: A foundation for computation - Martin (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Theorem 3.4 (Martin [8] For a space X maxD embedded as a G ffi subset of a Scott domain D, the notions of paracompactness, metrizability and complete metrizability are all equivalent.
		Paracompactness is that topological idea which allows us to extend the local to the global within a....
		Paracompactness is that topological idea which allows us to extend the local to the global within a topological space.
	citeseer.ist.psu.edu /context/1761451/0 (2426 words)

	iqexpand.com (Site not responding. Last check: 2007-11-05)
		Look for Paracompact hausdorff space in the Commons, our repository for free images, music, sound, and video.
		On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
		Click link for more info and facts about paracompact) paracompact Hausdorff space every open cover admits a (Click link for more info and facts about partition of unity) partition of unity subordinate...
	paracompact_hausdorff_space.iqexpand.com (402 words)

	American Mathematical Monthly: October, 1997
		Existence of partitions of unity for metric spaces is usually proved using the (equivalent) concept of paracompactness.
		Although the standard proof of paracompactness of metric spaces is the one given by M.E. Rudin, in his 1965 PhD thesis, Michael Mather showed that it is easier, for metric spaces, to show directly the existence of locally finite partitions of unity for arbitrary covers.
		We adapt Mather's argument to prove existence of a partition of unity subordinated to a countable open cover of a metric space.
	www.maa.org /pubs/monthly_oct97_toc.html (759 words)

	On Paracompactness of Metrizable Spaces - Borys, metrizable, of, Mathematics, mizar, JFM, pcomps, The, have ... (Site not responding. Last check: 2007-11-05)
		On Paracompactness of Metrizable Spaces - Borys, metrizable, of, Mathematics, mizar, JFM, pcomps, The, have (ResearchIndex)
		On Paracompactness of Metrizable Spaces - Leszek Borys Warsaw (1991)
		1 on paracompactness of metrizable spaces - Borys, metrizable et al.
	citeseer.ist.psu.edu /362617.html (529 words)

	PlanetMath: manifold
		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, topological manifold, Lagrange multipliers on manifolds
		Cross-references: between, mapping, path, differentiable, composition, restricted, representation, continuous function, homeomorphism, union, isomorphic, real analytic, class, continuously differentiable, Variables, real, functions, cover, domains, collection, injection, continuous, information, structure, open subsets, locally homeomorphic, separated, line, paracompactness, long line, definition of a manifold, metrizable, paracompact, equivalent, connected, topological space, Hausdorff, second countable, topological manifold, properties, coordinates
	www.planetmath.org /encyclopedia/CoordinateChart.html (332 words)

	On Pairwise Paracompactness (Site not responding. Last check: 2007-11-05)
		This paper answers a recent question concerning the relationship between two notions of paracompactness for bitopological spaces.
		Romaguera and Marin defined pairwise paracompactness in terms of pair open covers, motivated by a characterization of paracompactness due to Junnila.
		-pairwise paracompactness, and that the converse is false.
	anziamj.austms.org.au /JAMSA/V53/Part2/Ganster.html (86 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In this paper, a survey on various kinds of fuzzy paracompactness is presented.
		Firstly, we consider the concept of fuzzy paracompactness in the sense of Malghan and Benchalli (r-paracompactness,
		-paracompactness), thirdly, we consider the concept of fuzzy paracompactness in the sense of Abd El-Monsef, Zeyada, El-Deeb and Hanafy (fuzzy paracompactness, -fuzzy paracompactness) and finally the concepts of -paracompactness and fuzzy paracompactness on L-fuzzy topological space introduced by Yixiang and Shi and Zheng, respectively, are surveyed.
	www.pphmj.com /abstracts/fjms/vol13issue1/AB-11.htm (112 words)

	AMS Journals :: Print and Electronic
		Monotonically countably paracompact, collectionwise Hausdorff spaces and measurable cardinals.
		Characterizations of paracompactness and Lindelöfness by the separation property.
		A note on paracompactness in generalized ordered spaces.
	www.ams.org /joursearch/servlet/DoSearch?f1=msc&v1=54D20 (165 words)

	Atlas: On relative countable paracompactness by Yoshikazu Yasui (Site not responding. Last check: 2007-11-05)
		Some results on relative normality and relative paracompactness were obtained by I.Ju.
		In this talk, we shall discuss the relative version of countable paracompactness and study their characterizations etc..
		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 # caby-18.
	atlas-conferences.com /c/a/b/y/18.htm (215 words)

	Rudin (Site not responding. Last check: 2007-11-05)
		Also in 1952 the paper Concerning a problem of Souslin's continued her examination of the implications of R L Moore's axiom systems, this time motivated by a 1920 problem due to Souslin.
		Having looked at some of her earliest papers let us note that she is best known for her ability to construct counter-examples.
		Invited to be the Emmy Noether Lecturer for the Association for Women in Mathematics, she lectured on Paracompactness.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Rudin.html (1764 words)

	Yukinobu Yajima (Site not responding. Last check: 2007-11-05)
		Let X be a paracompact space and Y a stratifiable space.
		Then X× Y is paracompact if and only if X× Y is countably paracompact.
		Let X be a countably paracompact space and Y the closed image of a p-mosaic space.
	www.utm.edu /staff/jschomme/topology/c/a/a/j/121.htm (150 words)

	On Paracompactness of Metrizable Spaces
		The aim is to prove, using Mizar System, one of the most important result in general topology, namely the Stone Theorem on paracompactness of metrizable spaces [18].
		The remaining material is devoted to concepts and certain properties needed for the formulation and the proof of that theorem (see also [4]).
		A new proof that metric spaces are paracompact.
	mizar.uwb.edu.pl /JFM/Vol4/pcomps_2.html (170 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		One theme is normality and countable paracompactness versus paracompactness in locally compact spaces.
		Several longstanding problems in this area remain unsettled, e.g., the Arhangel'skii-Tall problem (Are normal locally compact metacompact spaces paracompact?) and Watson's problem (Is there a perfectly normal locally compact non- paracompact space in ZFC?) Another problem has led to interesting combinatorics involving ladder systems that will likely have further applications.
		Other problems to be investigated include the existence of Dowker filters, and questions of Watson and Cook involving connectedness.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9102725.txt (165 words)

	Tensor Analysis on Manifolds - Richard L Bishop, Samuel I. Goldberg Books - Lake Country Shop
		As for the topology needed to study differentiable manifolds, it is developed in the beginnning, though its not the best "quick untro to topology" Ive seen.
		Of course you can skip some of the sections such as Paracompactness.
		The only consequence is that you might not be able to follow some of the proofs later on.
	www.lakecountryshop.com /shop-item_id-0486640396-search_type-AsinSearch-locale-us.html (274 words)

	Artico-Marconi-Moresco references
		Umberto Marconi, On the Uniform Paracompactness of the Product of Two Uniform Spaces, Rend.
		Umberto Marconi, On Uniform Paracompactness of the $\omega_\mu$-metric spaces, Atti Acc.
		Umberto Marconi, Uniform Paracompactness and Uniform Weight, Rend.
	www.math.unipd.it /~artico/AMM.html (740 words)

	PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.), Vol. 55(69), pp. 98--104, 1994 (Site not responding. Last check: 2007-11-05)
		Abstract: Some properties of paracompactness in spaces which are not necessarily regular and some properties of almost closed mappings are studied.
		It is shown that some properties of paracompactness of regular spaces are hold even if the space is not necessarily regular.
		Beside that, additional conditions are given for almost closed mappings to have a closed graph.
	www.univie.ac.at /EMIS/journals/PIMB/069/12.html (99 words)

	On properties of relative metacompactness and paracompactness type by Elise M. Grabner, Gary C. Grabner and Kazumi ... (Site not responding. Last check: 2007-11-05)
		On properties of relative metacompactness and paracompactness type by Elise M. Grabner, Gary C. Grabner and Kazumi Miyazaki
		On properties of relative metacompactness and paracompactness type
		We study several natural relative properties of metacompactness and paracompactness types and the relationships among them.
	at.yorku.ca /b/a/a/k/71.htm (183 words)

	Matches for: Author=marconi, u* (Site not responding. Last check: 2007-11-05)
		[15] MR0778348 (86f:54037) Marconi, Umberto On the uniform paracompactness.
		[17] MR0780810 (86g:54035) Marconi, Umberto On uniform paracompactness of the $\omega\sb µ$-metric spaces.
		[18] MR0717001 (85c:54040) Marconi, Umberto On the uniform paracompactness of the product of two uniform spaces.
	www.math.unipd.it /~artico/reviews/marconi.htm (340 words)

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