Projective linear group - Revision history

en>Nbarth: /* Action on projective line */ anharmonic

2014-07-13T22:33:54Z

Action on projective line: anharmonic

← Older revision		Revision as of 00:33, 14 July 2014
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	~~{{Distinguish\|cofinality}}~~		Hi! I am Dalton. Acting often is a thing that I appreciate totally addicted to. My your own house is now in Vermont and I don't idea on changing it. I am a cashier. I'm not okay at webdesign but you can want to check brand new website: http://circuspartypanama.com<br><br>My webpage ... [http://circuspartypanama.com clash of clans hack password]
	In [[mathematics]], a '''cofinite''' [[subset]] of a set ''X'' is a subset ''A'' whose [[complement (set theory)\|complement]] in ''X'' is a finite set. In other words, ''A'' contains all but finitely many elements of ''X''. If the complement is not finite, but it is countable, then one says the set is [[cocountable]].

	~~These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the [[#Product topology\|product topology]] or [[#Direct sum\|direct sum]].~~

	~~==Boolean algebras==~~

	The set of all subsets of ''X'' that are either finite or cofinite forms a [[Boolean algebra (structure)\|Boolean algebra]], i.e., it is closed under the operations of union, intersection, and complementation. ~~This Boolean algebra~~ is ~~the '''finite-cofinite algebra''' on ''X''. A Boolean algebra ''A'' has~~ a ~~unique non-principal [[ultrafilter]] (i.e. a [[maximal filter]] not generated by a single element of the algebra) if and only if there is an infinite set ''X'' such~~ that ~~''A'' is isomorphic~~ to ~~the finite-cofinite algebra on ''X''~~. ~~In this case, the non-principal ultrafilter~~ is ~~the set of all cofinite sets.~~

	~~==Cofinite topology==~~

	~~The '''cofinite topology''' (sometimes called the '''finite complement topology''') is a [[topological space\|topology]] which can be defined on every set ''X''. It has precisely the [[empty set]]~~ and ~~all [[cofinite subset]]s of~~ '~~'X'' as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of ''X''. Symbolically, one writes the topology as~~

	~~:<math>\mathcal{T} = \{A \subseteq X \mid A=\varnothing \mbox{ or } X \setminus A \mbox{ is finite} \}</math>~~

	~~This topology occurs naturally in the context of the [[Zariski topology]]. Since [[polynomial]]s over a [[field (mathematics)\|field]] ''K'' are zero~~ on finite sets, or the whole of ''K'', the Zariski topology on ''K'' (considered as ''affine line'') is the cofinite topology. The same is true for any ''[[Irreducible component\|irreducible]]'' [[algebraic curve]]; it ~~is not true, for example, for ''XY'' = 0 in the plane~~.

	~~===Properties===~~
	* Subspaces: Every [[subspace topology]] of the cofinite topology is also a ~~cofinite topology~~.
	* Compactness: Since every [[open set]] contains all but finitely many points of '~~'X'', the space ''X'' is [[compact set\|compact]] and [[sequentially compact]].~~
	* Separation: The cofinite topology is the [[Comparison of topologies\|coarsest topology]] satisfying the [[T1 space\|T<sub>1</sub> axiom]]; i.e. it is the smallest topology for which every [[singleton set]] is closed. In fact, an arbitrary topology on ''X'' satisfies the T<sub>1</sub> axiom if and only if it contains the cofinite topology. If ''X'' is finite then the cofinite topology is simply the [[discrete space\|discrete topology]]. If ''X'' is not finite, then this topology is not [[T2 space\|T<sub>2</sub>]], [[regular space\|regular]] or [[normal space\|normal]], since no two nonempty open sets are disjoint (i.e. it is [[hyperconnected space\|hyperconnected]]).

	~~===Double-pointed cofinite topology===~~
	The '''double-pointed cofinite topology''' is the cofinite topology with every point doubled; that is, it is the [[topological product]] of the cofinite topology with the [[indiscrete topology]]. It is not [[T0 space\|T<sub>0</sub>]] or [[T1 space\|T<sub>1</sub>]], since the points of the doublet are [[topologically indistinguishable]]. It is, however, [[R0 space\|R<sub>0</sub>]] since the topologically distinguishable points are separable.

	An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together. Let ''X'' be the set of integers, and let ''O''<sub>''A''</sub> be a subset of the integers whose complement is the set ''A''. Define a [[subbase]] of open sets ''G''<sub>''x''</sub> for any integer ''x'' to be ''G''<sub>''x''</sub> = ''O''<sub>{''x'', ''x''+1}</sub> if ''x'' is an [[even number]], and ''G''<sub>''x''</sub> = ''O''<sub>{''x''-1, ''x''}</sub> if ''x'' is odd. Then the [[basis (topology)\|basis]] sets of ''X'' are generated by finite intersections, that is, for finite ''A'', the open sets of the topology are

	:~~<math>U_A~~ :~~= \bigcap_{x \in A} G_x </math>~~

	~~The resulting space is not T<sub>0<~~/~~sub> (and hence not T<sub>1<~~/~~sub>), because the points ''x'' and ''x'' + 1 (for ''x'' even) are topologically indistinguishable~~. ~~The space is, however, a [[compact space]], since it is covered by a finite union of the ''U''~~<~~sub~~>A<~~/sub~~>.

	~~==Other examples==~~

	~~===Product topology===~~
	~~The [[product topology]] on a product of topological spaces <math>\prod X_i</math>~~
	~~has [[basis (topology)\|basis]] <math>\prod U_i</math> where <math>U_i \subset X_i</math> is open, and cofinitely many <math>U_i = X_i</math>~~.

	~~The analog (without requiring that cofinitely many are the whole space) is the [[box topology]]~~.

	~~===Direct sum===~~
	~~The elements of the~~ [~~[direct sum of modules]] <math>\bigoplus M_i<~~/~~math> are sequences <math>\alpha_i \in M_i<~~/~~math> where cofinitely many <math>\alpha_i = 0</math>.~~

	~~The analog (without requiring that cofinitely many are zero) is the [[direct product]].~~

	~~==References==~~
	*{{Citation \| last1=Steen \| first1=Lynn Arthur \| author1-link=Lynn Arthur Steen \| last2=Seebach \| first2=J. Arthur Jr. \| author2-link=J. Arthur Seebach, Jr. ~~\| title=[[Counterexamples in Topology]] \| origyear=1978 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=[[Dover Publications\|Dover]] reprint~~ of ~~1978 \| isbn=978-0-486-68735-3 \| mr=507446 \| year=1995}} ''(See example 18)''~~

	~~[[Category:Basic concepts in infinite set theory]]~~
	~~[[Category:General topology]~~]

en>Monkbot: /* Exceptional isomorphisms */Fix CS1 deprecated date parameter errors

2014-01-28T21:41:08Z

Exceptional isomorphisms: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 23:41, 28 January 2014
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			{{Distinguish\|cofinality}}
			In [[mathematics]], a '''cofinite''' [[subset]] of a set ''X'' is a subset ''A'' whose [[complement (set theory)\|complement]] in ''X'' is a finite set. In other words, ''A'' contains all but finitely many elements of ''X''. If the complement is not finite, but it is countable, then one says the set is [[cocountable]].

			These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the [[#Product topology\|product topology]] or [[#Direct sum\|direct sum]].

	~~Eusebio~~ is the ~~name families use~~ to ~~call us~~ and ~~I think~~ the ~~software sounds quite good when you say it~~. ~~Idaho~~ is ~~our birth install~~. ~~I worn to be unemployed except now I am~~ a ~~great cashier~~. ~~My excellent say~~ it's not ~~superior~~ for ~~me but something I love doing~~ is ~~definitely to bake~~ but I'~~m thinking~~ on ~~starting interesting things~~. I'm not ~~good at~~ [~~http:~~//~~search~~.~~Huffingtonpost~~.~~com~~/~~search?q~~=~~web~~+~~design&s_it~~=~~header_form_v1 web design~~] ~~but you might aspire to check my website~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~Check out my webpage;~~ [~~http:~~//~~prometeu~~.~~net clash~~ of ~~clans hacks~~]		==Boolean algebras==

			The set of all subsets of ''X'' that are either finite or cofinite forms a [[Boolean algebra (structure)\|Boolean algebra]], i.e., it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the '''finite-cofinite algebra''' on ''X''. A Boolean algebra ''A'' has a unique non-principal [[ultrafilter]] (i.e. a [[maximal filter]] not generated by a single element of the algebra) if and only if there is an infinite set ''X'' such that ''A'' is isomorphic to the finite-cofinite algebra on ''X''. In this case, the non-principal ultrafilter is the set of all cofinite sets.

			==Cofinite topology==

			The '''cofinite topology''' (sometimes called the '''finite complement topology''') is a [[topological space\|topology]] which can be defined on every set ''X''. It has precisely the [[empty set]] and all [[cofinite subset]]s of ''X'' as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of ''X''. Symbolically, one writes the topology as

			:<math>\mathcal{T} = \{A \subseteq X \mid A=\varnothing \mbox{ or } X \setminus A \mbox{ is finite} \}</math>

			This topology occurs naturally in the context of the [[Zariski topology]]. Since [[polynomial]]s over a [[field (mathematics)\|field]] ''K'' are zero on finite sets, or the whole of ''K'', the Zariski topology on ''K'' (considered as ''affine line'') is the cofinite topology. The same is true for any ''[[Irreducible component\|irreducible]]'' [[algebraic curve]]; it is not true, for example, for ''XY'' = 0 in the plane.

			===Properties===
			* Subspaces: Every [[subspace topology]] of the cofinite topology is also a cofinite topology.
			* Compactness: Since every [[open set]] contains all but finitely many points of ''X'', the space ''X'' is [[compact set\|compact]] and [[sequentially compact]].
			* Separation: The cofinite topology is the [[Comparison of topologies\|coarsest topology]] satisfying the [[T1 space\|T<sub>1</sub> axiom]]; i.e. it is the smallest topology for which every [[singleton set]] is closed. In fact, an arbitrary topology on ''X'' satisfies the T<sub>1</sub> axiom if and only if it contains the cofinite topology. If ''X'' is finite then the cofinite topology is simply the [[discrete space\|discrete topology]]. If ''X'' is not finite, then this topology is not [[T2 space\|T<sub>2</sub>]], [[regular space\|regular]] or [[normal space\|normal]], since no two nonempty open sets are disjoint (i.e. it is [[hyperconnected space\|hyperconnected]]).

			===Double-pointed cofinite topology===
			The '''double-pointed cofinite topology''' is the cofinite topology with every point doubled; that is, it is the [[topological product]] of the cofinite topology with the [[indiscrete topology]]. It is not [[T0 space\|T<sub>0</sub>]] or [[T1 space\|T<sub>1</sub>]], since the points of the doublet are [[topologically indistinguishable]]. It is, however, [[R0 space\|R<sub>0</sub>]] since the topologically distinguishable points are separable.

			An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together. Let ''X'' be the set of integers, and let ''O''<sub>''A''</sub> be a subset of the integers whose complement is the set ''A''. Define a [[subbase]] of open sets ''G''<sub>''x''</sub> for any integer ''x'' to be ''G''<sub>''x''</sub> = ''O''<sub>{''x'', ''x''+1}</sub> if ''x'' is an [[even number]], and ''G''<sub>''x''</sub> = ''O''<sub>{''x''-1, ''x''}</sub> if ''x'' is odd. Then the [[basis (topology)\|basis]] sets of ''X'' are generated by finite intersections, that is, for finite ''A'', the open sets of the topology are

			:<math>U_A := \bigcap_{x \in A} G_x </math>

			The resulting space is not T<sub>0</sub> (and hence not T<sub>1</sub>), because the points ''x'' and ''x'' + 1 (for ''x'' even) are topologically indistinguishable. The space is, however, a [[compact space]], since it is covered by a finite union of the ''U''<sub>A</sub>.

			==Other examples==

			===Product topology===
			The [[product topology]] on a product of topological spaces <math>\prod X_i</math>
			has [[basis (topology)\|basis]] <math>\prod U_i</math> where <math>U_i \subset X_i</math> is open, and cofinitely many <math>U_i = X_i</math>.

			The analog (without requiring that cofinitely many are the whole space) is the [[box topology]].

			===Direct sum===
			The elements of the [[direct sum of modules]] <math>\bigoplus M_i</math> are sequences <math>\alpha_i \in M_i</math> where cofinitely many <math>\alpha_i = 0</math>.

			The analog (without requiring that cofinitely many are zero) is the [[direct product]].

			==References==
			*{{Citation \| last1=Steen \| first1=Lynn Arthur \| author1-link=Lynn Arthur Steen \| last2=Seebach \| first2=J. Arthur Jr. \| author2-link=J. Arthur Seebach, Jr. \| title=[[Counterexamples in Topology]] \| origyear=1978 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=[[Dover Publications\|Dover]] reprint of 1978 \| isbn=978-0-486-68735-3 \| mr=507446 \| year=1995}} ''(See example 18)''

			[[Category:Basic concepts in infinite set theory]]
			[[Category:General topology]]

128.84.124.184: /* Exceptional isomorphisms */

2012-08-30T22:17:22Z

Exceptional isomorphisms

New page

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