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Topic: Covering space

	PlanetMath: covering space
		Covering maps are especially important in the study of Riemann surfaces; in this context, one sometimes discusses a generalized notion of covering map called a “ramified covering”; this allows one to replace a discrete set of the local homeomorphisms by maps that locally look like
		Covering maps are also generalized in algebraic geometry; there the corresponding notion is that of étale morphism.
		This is version 17 of covering space, born on 2001-11-16, modified 2005-08-08.
	planetmath.org /encyclopedia/CoveringSpace.html (271 words)

	PlanetMath: covering space
		Covering maps are especially important in the study of Riemann surfaces; in this context, one sometimes discusses a generalized notion of covering map called a “ramified covering”; this allows one to replace a discrete set of the local homeomorphisms by maps that locally look like
		Covering maps are also generalized in algebraic geometry; there the corresponding notion is that of étale morphism.
		This is version 17 of covering space, born on 2001-11-16, modified 2005-08-08.
	www.planetmath.org /encyclopedia/CoveringSpace.html (270 words)

	PlanetMath: universal covering space
		A universal covering space is a covering space
		But not every topological space has a universal cover.
		This is version 3 of universal covering space, born on 2002-02-02, modified 2003-06-18.
	planetmath.org /encyclopedia/UniversalCoveringSpace.html (145 words)

	[No title]
		A covering transformation is a homeomorphism of a covering space with itself that preserves the covering map.
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
		PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-10-08)
		In mathematics, a covering space is a topological space C which "covers" another space X by a surjective local homeomorphism p : C â’ X called a covering map.
		Covering spaces are studied in algebraic topology, but they have important applications in many other branches of mathematics including differential topology, the theory of topological groups, and the theory of Riemann surfaces.
		The theory of covering spaces in deeply intertwined with the study of fundamental groups.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=covering_space (1762 words)

	The Covering Space Representation of Flexagon Operation
		Such a covering space may be constructed as follows: Within a circle, construct three arcs tangent to each other at the circumference, as shown in figure 2.9a.
		Thus the covering space as a whole represents a hypothetical ``ultimate" flexagon, from which any other flexagon could be produced by deleting various sides.
		This device indicates that, as we perform the operation of rotating the flexagon or splitting it to find a new side, which operation we identify with a new reflection in the covering space, we enter a new area or subspace within the co-vering space.
	theory.lcs.mit.edu /~edemaine/flexagons/Conrad-Hartline-1962/node4.html (2128 words)

	Universal Covering Space
		Let s~ and d be covering spaces for s, such that d is connected, and let π(d) embedded in s be a subgroup of π(s~) embedded in s.
		Conversely, a morphism between covering spaces is a lift of the projection of the first up to the second.
		Given a space s, or a space s with a designated base point a, the covering spaces form a category.
	www.mathreference.com /at-cov,ucs.html (1247 words)

	Polycut Multiple Universes (Site not responding. Last check: 2007-10-08)
		To illustrate that, each of the example covering spaces comes with a soap film that may be inserted into the covering space, showing how it blocks all lines of sight to the other sheets.
		The term "covering space" comes from the fact that the ground the garage is built on is covered with multiple levels of parking.
		Covering spaces whose branch curve is an unknotted loop differ only in the number of sheets, i.e.
	www.susqu.edu /facstaff/b/brakke/polycut/polycut.htm (4841 words)

	PlanetMath: universal covering space
		The universal covering space has the following universal property: If is a based universal cover, then for any connected based cover
		Clearly, if a universal covering exists, it is unique up to unique isomorphism.
		This is version 3 of universal covering space, born on 2002-02-02, modified 2003-06-18.
	www.planetmath.org /encyclopedia/UniversalCover.html (144 words)

	Springer Online Reference Works
		Initially, uniform spaces were used as tools for the study of the topologies (generated by them) (similar to the way a metric on a metrizable space was often used for the study of the topological properties of the space).
		The topology of a metrizable uniform space is paracompact, by Stone's theorem.
		The construction of a metrizable uniform space that is not uniformly paracompact (i.e.
	eom.springer.de /u/u095250.htm (1278 words)

	Math 290B--C Material
		The covering space of a cell complex is a cell complex.
		Notice that the lens space is covered by the 3-sphere is an example of 5.10.
		The covering space of a topological group is again a topological group.
	www.math.ucsd.edu /~neldredg/topology (890 words)

	Covering Spaces
		A covering space is actually two spaces, the total space (upstairs) and the base space (downstairs), and a continuous projection that carries the total space onto the base space.
		Given a path in the base space, establish even one point in its preimage, and the rest of the path's preimage in the total space is determined.
		In the world of covering spaces, components and path components are synonymous, and connected and path connected are synonymous.
	www.mathreference.com /at-cov,intro.html (723 words)

	Scientific FrontLine / Research opens up new dimensions in space
		"The size and complexity of the current International Space Station is limited by the size of the space capsules which transport the steel pieces into space," said Dr Kondyurin, a Russian-born research fellow in the School of Physics.
		When combined in space with large pieces of inflatable textile, it solidifies, producing a stable structure solid enough to use in buildings.
		Once in space, it is inflated using gas and combines with the polymer resin to form a strong, cylindrical frame 100 meters in length with a 10-meter diameter," explained Dr Kondyurin.
	www.sflorg.com /spacenews/sn081006_02.html (364 words)

	January-February 2002: Dedicated Equipment Space
		The basic rule in 110-26(f)(1)(a) defined the dedicated space and the last sentence of that rule stated "No piping, ducts, or equipment foreign to the electrical installation shall be located in this zone." Wording in the exception to 110-26(f)(1)(a) was similar to the former exception in 384-4(a)(1) but added a new reference to 110-26(f)(1)(b).
		Provisions in "(b) Foreign Systems" of the 1999 NEC are "The space equal to the width and depth of the equipment shall be kept clear of foreign systems unless protection is provided to avoid damage from condensation, leaks, or breaks in such foreign systems.
		The basic rule in (a) defines the space as to the structural ceiling or a maximum height of six feet, whichever is greater.
	www.iaei.org /subscriber/magazine/02_a/cox1.htm (1181 words)

	Covering Spaces and G Sets - 0506Topology
		There are similar theorems (call them theorem 1' and theorem 2') relating the category of based covering spaces with the category of based G-sets.
		If B is semi-locally simply connected, there exists a unique (up to base-point-preserving isomorphism) "universal covering space U of B" (a connected and simply connected covering U).
		Use the construction of U or the general lifting property for covering spaces to show that there is a left action of G on U.
	katlas.math.toronto.edu /0506-Topology/index.php?title=Covering_Spaces_and_G_Sets (662 words)

	Extraordinary Travel Requires Extraordinary Outfit - Space - RedOrbit (Site not responding. Last check: 2007-10-08)
		A space suit is like a blanket of Earth that completely surrounds an astronaut.
		"A space suit is equipped with both air conditioning and heating to maintain an even temperature of the astronaut," Gyuk said.
		Space suits are piped with gas to create air pressure.
	www.redorbit.com /news/space/775577/extraordinary_travel_requires_extraordinary_outfit/index.html?source=r_space (564 words)

	Topology MAT 530
		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.
		This covering is used to prove that the fundamental group of the circle is freely generated by a single element (i.e., it is isomorphic to the additive group of all integers)
		The source space of the covering is called the covering space, and the target is called the base space.
	www.math.sunysb.edu /~timorin/mat530.html (2896 words)

	SPACE.com -- Making History: China Launches its First Piloted Spaceflight
		China appears to be sharpening its war fighting space skills, from creating anti-satellite weaponry, building new classes of heavy-lift and small boosters, as well as improving an array of military space systems.
		Chinese space officials are pleased with the performance of their unpiloted Shenzhou 4 spacecraft.
		Chinese space officials are reportedly ready to loft an unpiloted Shenzhou IV spacecraft - a key test prior to flying a crew next year.
	www.space.com /missionlaunches/shenzhou_china_archive.html (1466 words)

	HOW THE PETRI-NETS ARE DERIVED FROM TREE-GENERATED ORDER . CONCEQUENCES TO THEIR IMPLEMENTATION .
		If B has a universal covering space E, then E is uniquely determined up to homeomorphism (provided B is locally path connected).
		A sufficient condition for the existence of universal covering space is given by the next theorem Munkress, (1975).
		A universal covering tree of a graph G, covers any other covering graph of G. Remark 15 The flow-charts of programs are finite connected graphs with two pointed vertices called the beginning and the end and with all their branches being oriented.
	www.softlab.ntua.gr /~kyritsis/PapersInComputerScience/Hrm98Stf.htm (1683 words)

	American Journalism Review
		The scramble to cover the February 1 loss of the Columbia highlighted what reporters have been noticing for years--the small army of press that covered NASA and the space program in the 1960s has dwindled to next to nothing.
		Space coverage among the television networks and radio is even spottier.
		On the networks, CBS space consultant Bill Harwood and NBC's 69-year-old space correspondent Jay Barbree were praised for their knowledgeable coverage on the morning of the Columbia disaster, along with CNN's Miles O'Brien.
	www.ajr.org /Article.asp?id=2813 (1406 words)

	Lost in space no more
		To help space enthusiasts keep up with the latest, Springer and Praxis Publishing are launching an annual update on recent space launches, missions and results, to be published every autumn.
		Providing a balanced and entertaining review of all the world's major space programmes, the annuals will cover space exploration from a variety of angles: looking back at past missions, reviewing those currently under way and looking to those planned for the future.
		Brian Harvey is an internationally respected author of books on space exploration, specialising in the history and evolution of the Russian space programme, and an acknowledged expert on the space programmes of Europe, China, Japan and India.
	www.eurekalert.org /pub_releases/2006-10/s-lis103006.php (348 words)

	CNS - Space: Italy: Arms Control
		The Union is actively engaged in civilian space programmes and depends increasingly on outer space for its economic and industrial development as well as for its security.
		The Union is actively engaged in civilian space programs and depends increasingly on outer space for its economic and industrial development as well as for its security.
		Outer space is not a new issue: scholars have enumerated at least seven principal treaties covering space and six international fora addressing space issues (this does not include the EU structures).
	cns.miis.edu /research/space/italy/arms.htm (756 words)

	CNS - Space: Italy: Arms Control
		The Union is actively engaged in civilian space programmes and depends increasingly on outer space for its economic and industrial development as well as for its security.
		The Union is actively engaged in civilian space programs and depends increasingly on outer space for its economic and industrial development as well as for its security.
		Outer space is not a new issue: scholars have enumerated at least seven principal treaties covering space and six international fora addressing space issues (this does not include the EU structures).
	www.cns.miis.edu /research/space/italy/arms.htm (756 words)

	SPACE.com --
		After one day in space, the two astronauts aboard China’s Shenzhou 6 spacecraft have already racked up a series of firsts for their country’s space program, not the least of which began with leaving their seats.
		China’s human spaceflight ambitions are surging forward, ranging from new spacesuits for a planned two-person space shot in 2005 to establishing a commercial space tourism industry in 20 years.
		She would work as a researcher at a planned Chinese space station, the reports said, adding that the pilots of China's spacecraft will still be men.
	www.space.com /missionlaunches/china_shenzhou6_archive.html (1345 words)

	Space Island Group - Space News
		He said about 50 students have gone through the space tourism course since it started in the winter of 1999 quarter.
		The school added the classes with the anticipation that commercial space travel and tourism were inevitable and coming soon.
		Yesterday, a California entrepreneur came to RIT to pitch his plan to have space stations filled with hotels and factories orbiting Earth by decade's end.
	www.spaceislandgroup.com /News/news5.html (309 words)

	Glossary (Site not responding. Last check: 2007-10-08)
		a topological space is compact if every collection of open sets that covers the space has a finite subset that also covers the space.
		The components of a topological space are its maximal connected subspaces.
		a separable space is one that has a countable dense subset, that is a countable subset whose closure is the whole space.
	mcraefamily.com /MathHelp/BasicSetTopologyGlossary.htm (2717 words)

	The following are preprint versions of papers (co)authored by Greg Conner in Adobe PDF format
		The main result is that the fundamental group of a second countable, connected, locally path connected one-dimensional metric space is free if and only if it is countable, if and only if the space has a universal cover.
		To begin, we define several homotopy theoretic conditions which we then prove are equivalent to the existence of a universal covering space.
		We use these equivalencies to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space.
	www.math.byu.edu /~conner/preprints/index.html (216 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A non-normal 3-fold covering space corresponds to this non-normal subgroup.
		The 3-fold covering space is also a Klein bottle.
		It would determine the group G of covering transformations of (C,c_0) over (S,s_0), since G is isomorphic to Norm(pi_1(C,c_0) in pi_1(S,s_0)) / pi_1(C,c_0) and that is independent of factoring out normal subgroups contained in pi_1(C,c_0).
	www.lehigh.edu /~dmd1/cov128 (282 words)

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