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Topic: Local compactness

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Compact space

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	Locally compact space - Wikipedia, the free encyclopedia
		Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.
		The notion of local compactness is important in the study of topological groups mainly because every locally compact Hausdorff group G carries natural measures called the Haar measures which allow one to integrate functions defined on G.
		The study of locally compact Abelian groups is the foundation of harmonic analysis, a field that has since spread to non-Abelian locally compact groups.
	www.wikipedia.org /wiki/Locally_compact_space (1326 words)

	Local field - Wikipedia, the free encyclopedia
		In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation.
		There is some inconsistency in usage, but usually a local field is further assumed to be locally compact, and often the field R of real numbers and the field C of complex numbers are considered to be local as well by virtue of their local compactness.
		The abelian Galois extensions of local fields are of particular interest and form the subject of local classfield theory.
	en.wikipedia.org /wiki/Local_field (598 words)

	local compactness (Site not responding. Last check: 2007-11-05)
		Locally compact Hausdorff spaces that are not compact
		Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X) using the Stone-Cech compactification.
		The Pontryagin dual of an abelian topological group A is locally compact iff A is locally compact.
	www.yourencyclopedia.net /Local_compactness.html (1332 words)

	FSCC State News Services: Summary of Recommendations for Revisions to Florida's Growth Management Program (Site not responding. Last check: 2007-11-05)
		Local comprehensive plans were now required to be consistent with the state comprehensive plan, also adopted in 1985, and the appropriate regional policy plan prepared by the regional planning councils to ensure that essential state and regional interests were considered.
		The local program should be integrated in the local comprehensive plan, including the establishment of thresholds that are flexible and appropriate to the size of the community and are responsive to regional variations.
		Local comprehensive plans should use a 2020 planning timeframe and consistent population projections to ensure the county aggregate population projections balance with the sum of all cities and unincorporated county projections.
	www.myflorida.com /fdi/fscc/news/state/9904/rev-gm.htm (4698 words)

	PlanetMath: examples of locally compact and not locally compact spaces
		All topological manifolds are locally compact since locally they look like Euclidean space.
		Any discrete space is locally compact, since the singletons can serve as compact neighborhoods.
		This is version 13 of examples of locally compact and not locally compact spaces, born on 2002-06-23, modified 2002-10-22.
	www.planetmath.org /encyclopedia/ExamplesOfLocallyCompactAndNotLocallyCompactSpaces.html (324 words)

	Local compactness (Site not responding. Last check: 2007-11-05)
		All open or closedsubsets of a locally compact Hausdorff space are locally compact in the subspace topology.
		The notion of local compactness is important in the study of topological groups mainly because every locally compact Hausdorff group G carries natural measures called the Haarmeasures which allow one to integrate functions defined on G.
		The study of locally compact Abelian groups is the foundationof harmonic analysis, a field that has since spread tonon-Abelian locally compact groups.
	www.therfcc.org /local-compactness-76312.html (1199 words)

	Local field (Site not responding. Last check: 2007-11-05)
		There issome inconsistency in usage, but usually a local field is further assumed to be locally compact, and often the field of real numbers and the field of complex numbers are considered to be local as well byvirtue of their local compactness.
		A local field of characteristic p can always be realized as the field of Laurent series in one variable withcoefficients in a finite field (also of characteristic p).
		As the quotient of consecutiveramification groups is abelian, Galois groups of local fields are always solvable.The abelian Galois extensions of local fields are of particular interest and form the subject of local classfield theory.
	www.therfcc.org /local-field-210870.html (547 words)

	PlanetMath: locally compact
		have a neighborhood which is actually compact, since compact open sets are fairly rare and the more relaxed condition turns out to be more useful in practice.
		However, it is true that a space is locally compact at
		This is version 2 of locally compact, born on 2002-05-15, modified 2004-10-24.
	www.planetmath.org /encyclopedia/LocallyCompact.html (100 words)

	Scottish Neighbourhood Statistics Data Zones Background Information: page 5 (Site not responding. Last check: 2007-11-05)
		Local knowledge is still important in evaluating the data zones created, and it is highly appropriate that people with extensive local knowledge should examine the zones and make suggestions for their improvement.
		Very compact data zones are in some cases difficult to construct, perhaps because the geography of social deprivation does not have a very compact pattern, or simply because the Output Areas, from which data zones are built up, may have highly elongated or convoluted shapes.
		We thought it was important to accept local councils’ suggestions where possible, on the grounds that the data zones should be as useful as possible to the community as a whole, and local authorities are likely to be among the most frequent users of Neighbourhood Statistics.
	www.scotland.gov.uk /library5/society/sndata-05.asp (8672 words)

	Local compactness (Site not responding. Last check: 2007-11-05)
		compactness local local movies local government local history local interest local intranet local movie local news local newspaper local phone local time local rock
		Local Government International Bureau Acts as the European arm of the Local Government Association for England and Wales, aiding local authorities to access European grants, promote local government concerns and influence policy.
		Local Sororities Directory of and resources for local sororities, including information on how to start a sorority, list of local-friendly retailers, and site-of-the-month.
	www.serebella.com /encyclopedia/article-Local_compactness.html (308 words)

	Local field - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-11-05)
		There is some inconsistency in usage, but usually a local field is further assumed to be locally_compact, and often the field of real_numbers and the field of complex_numbers are considered to be local as well by virtue of their local compactness.
		A local field of characteristic 0 is always a finite extension of the field Q''p'' of p-adic_numbers for some prime ''p''.
		A local field of characteristic ''p'' can always be realized as the field of Laurent series in one variable with coefficients in a finite_field (also of characteristic ''p'').
	www.indexsuche.com /Local_field.html (447 words)

	Relative Compactness from the Bitopological Point of View (Site not responding. Last check: 2007-11-05)
		The notion of relative compactness based on the relation of two topologies on the same set and used by Z. Balogh instead of the above-mentioned idea clearly reveals, even at first glance, the bitopological essence of this notion.
		Thus we are able to choose different kinds of bitopological local compactness leading to the relative compactness.
		Besides, the strengthening of relative compactness makes it possible to connect the resulting strong compactness with a special type local compactness by the equivalence relation.
	www.pmf.ukim.edu.mk /mathematics/dvalishvili.htm (282 words)

	Math 121 Course Information (Site not responding. Last check: 2007-11-05)
		Many of the issues addressed by topology, such as compactness of spaces and continuity of functions, are treated in a simpler setting in the analysis courses 131AB.
		The principal theorems are the Baire category theorem, the characterization of compact metric spaces, the theorem that continuous functions on a compact space are uniformly continuous, and the contraction mapping principle, which is perhaps the most important and useful tool in analysis.
		Topological spaces are introduced, along with the separation axioms and various notions as compactness, local compactness, connectedness, and path connectedness.
	www.math.ucla.edu /undergrad/courses/math121 (411 words)

	FuncAna
		Local compactness of finite dimensional normed spaces as a corollary of two facts:
		Local weak compactness of the dual spaces of normed vector spaces: closed bounded sets in the dual space are weakly compact.
		Local weak compactness of reflexive Banach spaces: closed bounded sets in a reflexive Banach space are weakly compact.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	Local Compactness Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-05)
		Looking For local compactness - Find local compactness and more at Lycos Search.
		Find local compactness - Your relevant result is a click away!
		Look for local compactness - Find local compactness at one of the best sites the Internet has to offer!
	www.artisticnudity.com /search/encyclopedia/Local_compactness (1463 words)

	Dipartimento di Matematica - Università di Torino (Site not responding. Last check: 2007-11-05)
		We give characterizations for a subspace of a hyperspace, endowed with either the Vietoris or Wijsman topology, to be compact or relatively compact.
		Then we characterize --- both globally and at the single points --- the local compactness of a hyperspace, endowed with either the Vietoris, or the Wijsman, or the Hausdorff metric topology.
		Using these general results, we also produce examples of hyperspaces which are locally compact exactly at the points with some special property.ely the Hilbert--Einstein Lagrangian.ariational inequality.
	www.dm.unito.it /quadernidipartimento/quaderni.php?action=view_abs&article=q38-00.htm (90 words)

	Compact space (Site not responding. Last check: 2007-11-05)
		One of the main reasons for studying compact spaces is because they are very nice generalisations of finite sets.
		For this reason, it is often said that "compactness is the next best thing to finiteness".
		For example, (0,1] is not compact, since the sequence (0,1/n] of closed sets (in (0,1]) is nested, and so clearly has the finite intersection property, but has empty intersection.
	www.portaljuice.com /compact_space.html (1285 words)

	Compactness and Uniform Continuity (Site not responding. Last check: 2007-11-05)
		Ordinary continuity of f is a local property, while uniform continuity is a global property since it says something about the behaviour of f over the whole space
		Since compactness allows us to pass from the local to the global, the next result is not surprising:
		Note Compactness is not a necessary condition on the domain for uniform continuity.
	at.yorku.ca /course/atlas2/node7.html (117 words)

	[No title]
		local lseq, x lseq := [] every x := !s do if type(x) == "list" then lseq := sflatten(x) else put(lseq, x) return lseq end procedure sground(seq, i) #: ground sequence to i local j /i := 1 j := smin !
		return lseq while j := get(x) do { lseq := sruns(i, j, 1) pull(lseq) i := j } put(lseq, i) return lseq end procedure sruns(xargs[]) # disconnected runs local lseq, i, j, k, limit, x1, x2, x3 if \node_gen then return node("sruns", xargs) x1 := copy(spromote(xargs[1])) x2 := copy(spromote(xargs[2])) x3 := copy(spromote(xargs[3]))
		end procedure sreplp(x1, x2) local lseq, i x1 := spromote(x1) x2 := spromote(x2) lseq := [] while i := get(x1) do every 1 to get(x2) do put(lseq, i) return lseq end procedure sundulant(x, sw) # get undulant local lseq, i, dir, cdir x := spromote(x) lseq := [x[1]]
	www.cs.arizona.edu /icon/library/src/procs/seqops.icn (844 words)

	Compact space - free-definition (Site not responding. Last check: 2007-11-05)
		A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence.
		The term "Compact" was introduced by Maurice René Fréchet in 1906.
		At one time, when primarily metric spaces were studied, compact was taken to mean the weaker sequentially compact (every sequence has a convergent subsequence).
	www.free-definition.com /Compact-space.html (1361 words)

	Local Compactness (Site not responding. Last check: 2007-11-05)
		is a compact neighbourhood of x!) but not compact.
		(iv) The set of rational numbers Q with its usual topology is not a locally compact space, for suppose otherwise; then 0 has a compact neighbourhood C in Q so we can choose
		Now J is closed in (compact) C and is therefore compact in R.
	at.yorku.ca /course/atlas2/node8.html (66 words)

	212 - Information (Site not responding. Last check: 2007-11-05)
		Continuous functions on compact spaces, their boundedness and attaining maxima and minima.
		Uniform continuity of continuous functions on compact spaces.
		Compactness in the space of continuous functions, Ascoli-Arzela theorem.
	www.maths.tcd.ie /~zaitsev/212.html (248 words)

	Directory of open access journals
		Under the assumption that $f$ satisfies super-linear and sub-critical growth conditions, we show that for small $epsilon$ there exist solutions that concentrate near local minima of $V$.
		The local minima may occur in unbounded components, as long as the Laplacian of $V$ achieves a strict local minimum along such a component.
		A penalization technique developed by Felmer and del~Pino is used to handle the lack of compactness and the absence of the Palais-Smale condition in the variational framework.
	www.doaj.org /abstract?id=89548&toc=y (166 words)

	Syllabi 2005-2006 B-KUL-G0P55A Topology
		Although the students should already have some experience with topological concepts such as open and closed sets, continuity, convergence, compactness and coherence from their previous education, this course makes it clear that topology and topological invariance can be introduced and studied in a general way and that there is also ample need for this.
		Previous education that contained basic concepts of mathematical analysis and of the study of sets is presupposed and/or recommended.
		When introducing a number of important topological invariants under the form of division properties, (local) coherence and (local) compactness, the students will be introduced to concepts and relevant properties that occur in many different disciplines of math.
	www.kuleuven.ac.be /onderwijs/aanbod/syllabi/G0P55AE.htm (570 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		f restricted to this compact closed ball is a homeomorhism (by compactness) onto its image.
		So it maps x to the interior of the image of the closed ball, hence to the interior of f(U).
		We need of R^n that its neighbourhoods are "closed ball-like", in having no retractions onto their boundaries, and local compactness as well.
	math.niu.edu /~rusin/known-math/98/invar_domain (258 words)

	Atlas: Local Compactness of the Hyperspace C(X) by Robbie Beane (Site not responding. Last check: 2007-11-05)
		Atlas: Local Compactness of the Hyperspace C(X) by Robbie Beane
		Given a metric space X, we analyze local compactness of the hyperspace C(X) of closed, connected subsets, with the goal of obtaining a characterization for local compactness of the hyperspace.
		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 # capc-55.
	atlas-conferences.com /cgi-bin/abstract/capc-55 (79 words)

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