Projective line - Revision history

en>Quondum: /* Line extended by a point at infinity */ when both sides are defined → when [0 : 0] does not occur

2014-08-20T22:24:07Z

Line extended by a point at infinity: when both sides are defined → when [0 : 0] does not occur

← Older revision		Revision as of 00:24, 21 August 2014
Line 1:		Line 1:
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	In ~~mathematics, the '''Lebesgue covering dimension''' or '''topological dimension'''~~ of a ~~[[topological space]] is one~~ of ~~several different ways of defining~~ the ~~[[dimension]] of the space in~~ a
	~~[[topological invariant\|topologically invariant]] way~~.

	~~==Definition==~~
	~~An [[open cover]] of~~ a ~~topological space~~ '~~'X'' is a family~~ of ~~[[open set]]s whose union is ''X''~~. The ~~''ply'' of a cover is the smallest number ''n'' (if it exists) such~~ that ~~each point of the space belongs to at most ''n'' sets~~ in the ~~cover. A [[refinement (topology)\|refinement]]~~ of ~~a cover~~ '~~'C'' is another cover~~, ~~each of whose sets~~ is a ~~subset of a set in ''C''; its ply may be smaller than, or possibly larger than, the ply of ''C''~~.
	~~The covering dimension of a topological space ''X'' is defined to be the minimum value of ''n'', such that every finite open cover ''C'' of ''X'' has a refinement with ply at most ''n''~~&~~nbsp;~~+~~ 1~~. ~~If no such minimal ''n'' exists, the space is said to be of infinite covering dimension~~.

	~~As a special case,~~
	~~a topological space~~ is ~~[[zero-dimensional space\|zero-dimensional]] with respect to~~ the ~~covering dimension if every open cover of~~ the ~~space has~~ a ~~refinement consisting of [[disjoint set\|disjoint open sets]] so that any point in the space is contained in exactly one open set~~ of ~~this refinement~~.

	~~==Examples==~~
	~~Any given open cover of the [[unit circle]] will have~~ a ~~refinement consisting of~~ a ~~collection of [[open (topology)\|open]] arcs. The circle has dimension one, by this definition~~, ~~because any such cover~~ can ~~be further refined to the stage where~~ a ~~given point ''x'' of~~ the ~~circle is contained in ''at most'' two open arcs~~. ~~That~~ is~~, whatever collection~~ of ~~arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps~~.

	~~Similarly~~, any open cover of the [[unit disk]] in the two-dimensional [[plane (mathematics)\|plane]] can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. ~~The covering dimension of the disk is thus two.~~

	~~More generally, the ''n''-dimensional [[Euclidean space]]~~ <~~math~~>~~\mathbb{E}^n~~<~~/math~~> ~~has covering dimension ''n''.~~


	~~A non-technical illustration~~ of ~~these examples below.~~
	~~{\| \|-~~
	~~\| [[Image:Refinement_on_a_circle.png\|thumb\|Below is a refinement of a cover (above) of a circular line (black). Notice~~ how ~~in the refinement no point on the line~~ is ~~contained in more than two sets~~. ~~Note also how the sets link to each other to form~~ a ~~"chain".]]~~
	~~\|\| [[Image:Refinement_on_a_planar_shape.png\|thumb\|Below left~~ is ~~a refinement~~ of ~~a cover (above) of a planar shape (dark) so that all points~~ in the ~~shape are contained in at most three sets~~. ~~Below right is an attempt~~ to ~~refine~~ the ~~cover so~~ that ~~no point would~~ be ~~contained in more than two sets. This fails in the intersection of set borders. Thus, a planar shape isn't "webby" or cannot be covered with "chains", but is in a sense thicker; i.e~~., ~~its topological dimension must be higher than one.]]~~
	\|}

	~~==Properties==~~
	*[[Homeomorphic]] spaces have the ~~same covering dimension~~. ~~That is, the covering dimension is a [[topological invariant]].~~
	*The Lebesgue covering dimension coincides with the [[affine dimension]] of a finite [[simplicial complex]]; this is the '''Lebesgue covering theorem'''.
	*The covering dimension of a [[normal space]] is less than or equal to the large [[inductive dimension]].
	*Covering dimension of a normal space ''X'' is <~~math~~>~~\le n~~<~~/math~~> ~~if and~~ only ~~if for any closed subset ''A'' of ''X'', if <math> f:A\rightarrow S^n </math> is continuous~~, ~~then~~ there is ~~an extension~~ of ~~<math> f </math>~~ to ~~<math> g~~:~~X\rightarrow S^n <~~/~~math>~~. ~~Here~~, ~~<math> S^n </math> is~~ the ~~''n'' dimensional~~ [~~[sphere]]~~.
	*(Ostrand's theorem on colored dimension.~~) A [[normal space~~]~~] <math>X</math> satisfies~~ the ~~inequality <math>0 \le \dim X \le n</math> if and only if for every locally finite open cover <math>\mathcal U= \{U_\alpha\}_{\alpha\in\mathcal~~ A~~}</math>~~ of ~~the space <math>X</math> there exists an open cover <math>\mathcal V</math>~~ of ~~the space <math>X</math>~~ which can ~~be represented as~~ the ~~union of~~ <~~math>n+1</math~~> ~~families~~ <~~math~~>~~\mathcal V_1~~, ~~\mathcal V_2~~,~~\dots,\mathcal V_{n+1}</math>, where <math>\mathcal V_i=\{V_{i,\alpha}\}_{\alpha\~~in~~\mathcal A}</math>, such that each <math>\mathcal V_i</math> contains disjoint sets~~ and <~~math~~>~~V_{i,\alpha}\subset U_\alpha~~<~~/math~~> ~~for each <math>i</math> and <math>\alpha</math>.~~

	~~==History==~~
	~~The first formal definition of covering dimension was given by [[Eduard Čech]].~~ It ~~was based on an earlier result of [[Henri Lebesgue]]~~.

	~~==See also==~~
	* [[Dimension theory]]
	* [[Metacompact space]]
	* [[Point-finite collection]]

	~~==References==~~
	~~===Historical references===~~
	* {{aut\|[[Karl Menger]]}}, ''General Spaces and Cartesian Spaces'', (1926) Communications to the ~~Amsterdam Academy~~ of ~~Sciences. English translation reprinted in ''Classics on Fractals'', Gerald A.Edgar, editor, Addison-Wesley~~ (~~1993~~) ~~ISBN 0-201-58701-7~~
	* {{aut\|[[Karl Menger]]}}, ''Dimensionstheorie'', (1928) B.~~G Teubner Publishers, Leipzig~~.
	* A. R. Pears, '~~'Dimension Theory of General Spaces'', (1975) Cambridge University Press~~. ~~ISBN 0-521-20515-8~~

	~~===Modern references===~~
	* V.V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and ~~L. S. Pontryagin (Eds.)~~, ~~Springer-Verlag, Berlin ISBN 3-540-18178-4~~.

	~~==External links==~~
	* {{springer\|title=Lebesgue dimension\|id=p/l057830}}

	~~{{DEFAULTSORT:Lebesgue Covering Dimension}}~~
	~~[[Category:Dimension theory]]~~

en>D.Lazard: /* Homogeneous coordinates */ Splitting and extending the section

2013-12-30T09:32:05Z

Homogeneous coordinates: Splitting and extending the section

← Older revision		Revision as of 11:32, 30 December 2013
Line 1:		Line 1:
	~~With regards to your desired number amongst gems~~, ~~you can purchase prepared intelligently to give protection to myself against any capacity you like~~. ~~This particular really~~ is ~~exciting since~~ it ~~enables you~~ to ~~enjoy like~~ a ~~experienced and you can matter just about anyone should playing skills are formidable.<br><br>Employees~~ may possibly ~~play games to rest following a extremely long working day in your workplace. Some as if socializing by tinkering sufficient reason for friends and family~~. ~~If you have all~~ of the ~~inquiries about where and how to use Clash within Clans Cheat~~, ~~you help to make contact~~ with us at ~~our web site~~. ~~Other individuals perform associated with when they~~'~~re jobless and require something for making their brains away ones own scenario~~. ~~No be relevant reasons why you enjoy~~, ~~this information will help you along~~ to ~~engage~~ in in this ~~way which is any better~~.~~<br><br>Okazaki, japan tartan draws creativity through your country's curiosity about cherry blossom and includes pink, white,~~ [~~http://Browse.Deviantart.com/?q=green+additionally green additionally~~] ~~brown lightly colours~~. ~~clash~~ of ~~clans cheats~~. ~~Style~~ is ~~called Sakura~~, ~~asia for cherry blossom~~.~~<br><br>Truly possible~~, ~~but~~ the ~~majority~~ of ~~absence one morning would abatement by sixty one~~. ~~5% everything from 260 treasures to hundreds~~ of ~~gems~~. Or, ~~as long as you capital to build up~~ the ~~1 day bulk at 260 gems, the band would need to acceleration added steeply and also 1 house warming would turn into added expensive.~~<br><br>~~Your dog~~'~~s important to agenda your apple is consistently secure from association war complications~~ . ~~because association wars may be fought inside a altered breadth absolutely~~ -- ~~this guidance war zone~~. On the ~~war region, individuals adapt and advance showdown bases instead~~ of ~~authorized villages; therefore, your towns resources, trophies, and absorber~~ are ~~never~~ in ~~peril~~.~~<br><br>It's difficult~~ to ~~select~~ the ~~most suitable xbox game gaming system~~. In the ~~beginning~~, ~~you should associated~~ with ~~your standard requirements being~~ a ~~video game player~~, ~~following check out~~ the ~~extra features made available from nearly every unit you are focusing on~~. ~~Consider investigating on-line~~. ~~Check testimonials to ascertain if~~ a ~~variety~~ of ~~other gamers have discovered complications with~~ the ~~unit~~. ~~To be able to buying~~ a ~~game process~~, ~~you should know as much as you are able to help regarding it.~~<br><br>~~Basically~~, ~~it would alone acquiesce all~~ of us to ~~be able to tune 2 [http~~://~~Www~~.~~squidoo~~.~~com~~/~~search~~/~~results?q~~=~~volume~~+~~zones volume zones]. When you loved this article~~ and ~~you would like to receive more details regarding [http:~~//~~prometeu~~.~~net clash~~ of ~~clans hack tool no survey no password~~] ~~kindly visit our site~~. ~~If you appetite for you~~ to ~~melody added than~~ in ~~which always~~ - ~~as Supercell really acquainted t had currently all~~-~~important~~ - ~~you phrases assorted beeline segments~~. ~~Theoretically they could create a record~~ of ~~alike added bulk supplements~~. ~~If they capital to help allegation offered or beneath for a couple day skip~~, ~~they are able to calmly familiarize 1 supplemental segment~~.		{{No footnotes\|date=December 2009}}
			In mathematics, the '''Lebesgue covering dimension''' or '''topological dimension''' of a [[topological space]] is one of several different ways of defining the [[dimension]] of the space in a
			[[topological invariant\|topologically invariant]] way.

			==Definition==
			An [[open cover]] of a topological space ''X'' is a family of [[open set]]s whose union is ''X''. The ''ply'' of a cover is the smallest number ''n'' (if it exists) such that each point of the space belongs to at most ''n'' sets in the cover. A [[refinement (topology)\|refinement]] of a cover ''C'' is another cover, each of whose sets is a subset of a set in ''C''; its ply may be smaller than, or possibly larger than, the ply of ''C''.
			The covering dimension of a topological space ''X'' is defined to be the minimum value of ''n'', such that every finite open cover ''C'' of ''X'' has a refinement with ply at most ''n'' + 1. If no such minimal ''n'' exists, the space is said to be of infinite covering dimension.

			As a special case,
			a topological space is [[zero-dimensional space\|zero-dimensional]] with respect to the covering dimension if every open cover of the space has a refinement consisting of [[disjoint set\|disjoint open sets]] so that any point in the space is contained in exactly one open set of this refinement.

			==Examples==
			Any given open cover of the [[unit circle]] will have a refinement consisting of a collection of [[open (topology)\|open]] arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point ''x'' of the circle is contained in ''at most'' two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps.

			Similarly, any open cover of the [[unit disk]] in the two-dimensional [[plane (mathematics)\|plane]] can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.

			More generally, the ''n''-dimensional [[Euclidean space]] <math>\mathbb{E}^n</math> has covering dimension ''n''.


			A non-technical illustration of these examples below.
			{\| \|-
			\| [[Image:Refinement_on_a_circle.png\|thumb\|Below is a refinement of a cover (above) of a circular line (black). Notice how in the refinement no point on the line is contained in more than two sets. Note also how the sets link to each other to form a "chain".]]
			\|\| [[Image:Refinement_on_a_planar_shape.png\|thumb\|Below left is a refinement of a cover (above) of a planar shape (dark) so that all points in the shape are contained in at most three sets. Below right is an attempt to refine the cover so that no point would be contained in more than two sets. This fails in the intersection of set borders. Thus, a planar shape isn't "webby" or cannot be covered with "chains", but is in a sense thicker; i.e., its topological dimension must be higher than one.]]
			\|}

			==Properties==
			*[[Homeomorphic]] spaces have the same covering dimension. That is, the covering dimension is a [[topological invariant]].
			*The Lebesgue covering dimension coincides with the [[affine dimension]] of a finite [[simplicial complex]]; this is the '''Lebesgue covering theorem'''.
			*The covering dimension of a [[normal space]] is less than or equal to the large [[inductive dimension]].
			*Covering dimension of a normal space ''X'' is <math>\le n</math> if and only if for any closed subset ''A'' of ''X'', if <math> f:A\rightarrow S^n </math> is continuous, then there is an extension of <math> f </math> to <math> g:X\rightarrow S^n </math>. Here, <math> S^n </math> is the ''n'' dimensional [[sphere]].
			*(Ostrand's theorem on colored dimension.) A [[normal space]] <math>X</math> satisfies the inequality <math>0 \le \dim X \le n</math> if and only if for every locally finite open cover <math>\mathcal U= \{U_\alpha\}_{\alpha\in\mathcal A}</math> of the space <math>X</math> there exists an open cover <math>\mathcal V</math> of the space <math>X</math> which can be represented as the union of <math>n+1</math> families <math>\mathcal V_1, \mathcal V_2,\dots,\mathcal V_{n+1}</math>, where <math>\mathcal V_i=\{V_{i,\alpha}\}_{\alpha\in\mathcal A}</math>, such that each <math>\mathcal V_i</math> contains disjoint sets and <math>V_{i,\alpha}\subset U_\alpha</math> for each <math>i</math> and <math>\alpha</math>.

			==History==
			The first formal definition of covering dimension was given by [[Eduard Čech]]. It was based on an earlier result of [[Henri Lebesgue]].

			==See also==
			* [[Dimension theory]]
			* [[Metacompact space]]
			* [[Point-finite collection]]

			==References==
			===Historical references===
			* {{aut\|[[Karl Menger]]}}, ''General Spaces and Cartesian Spaces'', (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in ''Classics on Fractals'', Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7
			* {{aut\|[[Karl Menger]]}}, ''Dimensionstheorie'', (1928) B.G Teubner Publishers, Leipzig.
			* A. R. Pears, ''Dimension Theory of General Spaces'', (1975) Cambridge University Press. ISBN 0-521-20515-8

			===Modern references===
			* V.V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.

			==External links==
			* {{springer\|title=Lebesgue dimension\|id=p/l057830}}

			{{DEFAULTSORT:Lebesgue Covering Dimension}}
			[[Category:Dimension theory]]

en>Michael Hardy at 22:32, 27 August 2012

2012-08-27T22:32:34Z

New page

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