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Topic: Compact space

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	Compact space - Wikipedia, the free encyclopedia
		For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).
		A compact subset of a Hausdorff space is closed.
		Countably compact spaces are pseudocompact and weakly countably compact.
	en.wikipedia.org /wiki/Compact_space (1380 words)

	PlanetMath: compact
		with its subspace topology is a compact topological space.
		Note: Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space.
		This is version 6 of compact, born on 2001-10-25, modified 2004-03-28.
	planetmath.org /encyclopedia/Compact.html (258 words)

	PlanetMath: closed set in a compact space is compact
		is compact, we show that an arbitrary open cover has a finite subcover.
		"closed set in a compact space is compact" is owned by mathcam.
		This is version 6 of closed set in a compact space is compact, born on 2003-04-11, modified 2003-06-19.
	planetmath.org /encyclopedia/AClosedSetInACompactSpaceIsCompact.html (105 words)

	Staff Report - April 2, 2002 - Agenda Item - 02
		The per space construction cost would therefore be between $13,860 and $20,460 for a standard space and between $11,340 and $16,740 for a compact space.
		Since general office space must be parked at a rate of three spaces per 1,000 square feet of floor area, the cost of providing all full-size spaces in lieu of 45 percent compact spaces is estimated at $3,402 to $5,022 per 1,000 square feet of floor area.
		Compact spaces are not intended to accommodate the pickup trucks and SUVs that are common on the road today and frequently parked in compact spaces.
	www.ci.burbank.ca.us /agendas/ag_council/2005/sr061405_8.html (3129 words)

	Algebraic Topology: Topology
		A topological space is a set X together with a collection of subsets OS the members of which are called open, with the property that (i) the union of an arbitrary collection of open sets is open, and (ii) the intersection of a finite collection of open sets is open.
		A topological space is called metric when there is a distance function determining the topology (i.e., open balls for the metric are open sets, and conversely, if a point x lies in an open set U then for some positive e the ball with radius e around x is contained in U.
		A Hausdorff space X is normal if and only if for each pair of disjoint closed sets A and B there exists a map f from X to the unit interval I that is identically 0 on A and identically 1 on B.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	The Compact of Cape Cod Conservation Trusts, Inc. HOME
		The Compact works with 25 local and regional land trust and watershed associations on their projects to acquire and manage important natural areas as protected open space.
		The Compact of Cape Cod Conservation Trusts believes there is still an opportunity to preserve the essence of Cape Cod which exists in the collective imagination: a place of white sand beaches, broad salt marshes, quiet pine woods, intriguing cranberry bogs, and startlingly blue kettle ponds.
		The effort of the land trusts of The Compact is a significant supplement to the work of government in protecting open space, but it is not a substitute.
	www.compact.cape.com (500 words)

	Summary
		A topological space X is called compact if every open covering of X can be reduced to a finite sub-covering.
		The interval [0, 1] in R (with its usual topology) is compact.
		A closed subset of a compact space is compact (in the subspace topology).
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/Summary.html (953 words)

	Stone-Cech Compactification - NoiseFactory Science Archives (http://noisefactory.co.uk)
		A space is compact provided every open cover has a subcover containing only finitely many members (a finite subcover).
		Compact spaces are very popular with topologists, because they're very easy to reason about.
		If X is one of the standard spaces used in complicated mathematics it probably isn't compact, and that means it may be quite hard to reason about its properties.
	noisefactory.co.uk /maths/stone-cech.html (1649 words)

	Atlas: On Weak Reflections in Some Classes of Topological Spaces by Martin Maria Kovar
		A space has a weak reflection in compact spaces iff its Wallman remainder is finite.
		Husek proved that the class of compact spaces is not weakly reflective in topological spaces but it is natural to study the weak reflections "below" as well as "above" compactness.
		It should be noted that by a locally compact space we mean a space in which each point has a closed compact neighborhood.
	atlas-conferences.com /cgi-bin/abstract/caeh-44 (910 words)

	Space for Compact Mailboxes (Site not responding. Last check: 2007-10-21)
		Although it is referred to as "Compacting mailboxes", it actually duplicates the mailboxes to compress it first.
		To successfully Compact Mailboxes in Eudora, you will need to have free space on the disk equal to the size of the mailbox you are trying to compact.
		If you are compacting all of your mailboxes, you will need to have free space equal to the size of your largest mailbox.
	www.cit.cornell.edu /helpdesk/mac/email/compactspace.html (103 words)

	Compact and Hausdorff
		An important corollary is that a continuous map of a compact space into a hausdorff space is bicontinuous.
		If a continuous function is invertable, and maps a compact space onto a hausdorff space, the map is bicontinuous, and implements a homeomorphism.
		We already showed that a map from a compact space onto a hausdorff space is a homeomorphism, and that isn't the case here; hence the domain of f is not compact.
	www.mathreference.com /top-cs,haus.html (693 words)

	AMCA: Finite Approximation of Compact Hausdorff Spaces by R.G. Wilson
		However, our goal here is not simply to study finite spaces, but to use them to approximate other (infinite) topological spaces.
		space is finitely approximable if and only if it is compact.
		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/a/o/86.htm (546 words)

	Locally Compact (Site not responding. Last check: 2007-10-21)
		A space is locally compact if any point x has an open neighborhood q about it whose closure is compact.
		The space s is locally compact iff the open sets with compact closures form a base.
		A closed subspace of a compact set is compact, so the intersection of two base sets is another base set.
	www.mathreference.com /top-cs,locc.html (276 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Thus for subsets of a _complete_ metric space, X is compact iff it is closed and _totally_ bounded.
		For spaces which are not metric spaces, (a) makes perfect sense, (d) makes no sense, and (b) and (c) have to be defined appropriately.
		Date: 28 Jan 1995 19:28:26 GMT A metric space is compact iff it is complete and totally bounded.
	www.math.niu.edu /~rusin/papers/known-math/95/compact.nss (1036 words)

	Exercises 9
		Give an example of a subspace of a metric space which is closed and bounded but not compact.
		Prove that any compact subset of a metric space is closed and bounded.
		H is a continuous bijection from a compact space C to a Hausdorff space H prove that f is a homeomorphism.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Tutorials/T9.html (293 words)

	4. a compact universe
		Given derivation from the study of continuity, topology concerns itself with things which are close together while disregarding those which are far apart.
		we can bound the metric of a space while keeping the topology by changing the metric to a new metric, the minimum of 1 and the old distance between two points.
		to be contained (or embedded) in a compact space it is necessary and sufficient that for each pair consisting of a closed set
	www.math.buffalo.edu /~sww/classes/COMPACT/COMPACT4.html (244 words)

	A.P. Kombarov (Site not responding. Last check: 2007-10-21)
		The exponential space \exp(X) is the set of all non-empty closed subsets of X with Vietoris (finite) topology.
		We note here that a space X is a metrizable compact space if \exp(X) is hereditarily normal [3] or is regular hereditarily countably paracompact [5].
		If X is a countably compact space and if \exp(X) is hereditarily weakly normal, then X is a perfectly normal hereditarily separable compact space.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/55.htm (697 words)

	BGU Set Theory and Topology Seminars
		Abstract: A flow is, by the definition, an action of G on a topological space X, where G is either the group of integers or the group of the real numbers; we add the adjective discrete or continuous to specify the group.
		An interesting problem is which topological spaces admit minimal flows (a flow is minimal if all its orbits are dense).
		Finally, I will prove that if a Hausdorff space X admits a flow whose all forward orbits are dense then X is either compact or else X is nowhere locally compact.
	www.math.bgu.ac.il /~arkady/topologyseminar/topologyseminararchive.htm (543 words)

	Squeeze Is On King-Size Cars That Hog Tiny Parking Spots / Palo Alto to ticket compact space violators
		In general, compact stalls are a remnant of the 1970s, when gas shortages prompted the manufacture of smaller, more fuel-efficient cars.
		Parking lots were designed with compact spaces to fit more cars and to spur the public to buy smaller cars to fit into the spaces.
		When she was looking for a place to park her Honda Prelude, several large cars were parked in compact spots that she said she could have parked in.
	www.sfgate.com /cgi-bin/article.cgi?file=/chronicle/archive/1999/08/02/MN55782.DTL (1367 words)

	Ivan Gotchev (Site not responding. Last check: 2007-10-21)
		A topological space X is s -compact [2] if every sequentially open cover of X has a finite subcover.
		There exists an example of a compact and sequentially compact space, which is not s -compact [2].
		A topological space X is called irreducible [1] if the intersection of any finite number of open nonempty sets in X is nonempty.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/24.htm (452 words)

	Compact Space
		The Compact is a loose association of nominally seven sentient species.
		For these reasons, other Compact Space citizens just keep out of their way and try not to get their attention.
		The rest of the Compact considers getting even that much of the idea of trade across to the knnn to be a victory of sorts.
	www.chebucto.ns.ca /~af380/CompactSpace.html (1029 words)

	Compact Automation Products
		CAP is a market leading manufacturer of space efficient custom automation solutions and Standard pneumatic automation components.
		Compact has over 25 patents and has manufactured over 50,000 modified and custom designs.
		From simple porting changes to 100% customization, we engineer to your unique design requirements.
	www.compactair.com (45 words)

	Sequential compactness (Site not responding. Last check: 2007-10-21)
		For metric spaces there is another, perhaps more natural way of thinking about compactness.
		A metric space is sequentially compact if every bounded infinite set has a limit point.
		In fact, a metric space is compact if and only if it is sequentially compact.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L22.html (217 words)

	Introduction
		4], a new approach to proving these kinds of results was given, providing a transference principle for spaces of measures.
		In that paper, the action was from a locally compact abelian group into a space of isomorphisms on the space of measures of a sigma algebra.
		A primary requirement that the action had to satisfy was what was called sup path attaining, a property that was satisfied, for example, by the setting of Forelli (Baire measures on a locally compact topological space).
	www.math.missouri.edu /~stephen/preprints/helson-lowdenslager5/node1.html (942 words)

	Compact Hausdorff space with continuous function (Site not responding. Last check: 2007-10-21)
		is automatically compact (since every closed subset of a Hausdorff space is compact).
		is continuous it maps closed compact sets to closed compact sets.
		maybe also use the fact that a continuous image of a compact space is compact.
	www.physicsforums.com /showthread.php?p=788776#post788776 (1103 words)

	Boing Boing: Micro-Compact Homes inspired by first class air travel (Site not responding. Last check: 2007-10-21)
		These "Micro-Compact Homes" were designed at the Technical University of Munich and the Tokyo Institute of Technoloogy, inspired by the highly designed compact spaces in first-class airplane cabins and Smart cars.
		The tiny cube provides a double bed on an upper level and working table and dining space for four or five people on a lower level.
		The entrance lobby has triple use and functions as a bathroom and drying space for clothing.
	www.boingboing.net /2005/08/20/microcompact_homes_i.html (221 words)

	Decorating : Small Space : Compact Kitchen : Home & Garden Television
		With the help of designer Dee David, Mavin gutted the kitchen and introduced light colors, beautiful tile and, by using compact appliances, a dishwasher.
		A three-tier lazy Susan over the sink and pull-out drawers made what little storage space there is more accessible and functional.
		A stacked washer and dryer in the corner is efficient, compact and functional.
	www.hgtv.com /hgtv/dc_design_small_space/article/0,1793,HGTV_3382_1396256,00.html (279 words)

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