Distributed coordination function - Revision history

192.195.83.201 at 00:10, 22 November 2014

2014-11-22T00:10:20Z

← Older revision		Revision as of 02:10, 22 November 2014
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2014-02-28T16:08:42Z

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← Older revision		Revision as of 18:08, 28 February 2014
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	In [~~[topology]~~] and ~~related areas of~~ [~~[mathematics]], the set of all possible topologies on a given set forms a [[partially ordered set]]~~. ~~This [[order relation]~~] ~~can be used for '''comparison of the topologies'''~~.		My person who wrote some [http://Thesaurus.com/browse/article article] is called Leland but it's not those most masucline name presently. Managing people is very much where his primary income comes from. His wife and him live back in Massachusetts and he has everything that he needs there. Base jumping is something that she has been doing for many. He might be running and maintaining another blog here: http://[http://www.google.de/search?q=circuspartypanama circuspartypanama].com<br><br>

	~~== Definition ==~~		Have a look at my site ... how to hack clash of clans; [http://circuspartypanama.com circuspartypanama.com],

	~~Let τ<sub>1</sub> and τ<sub>2</sub> be two topologies on~~ a ~~set ''X'' such that τ<sub>1</sub> is [[subset\|contained in]] τ<sub>2</sub>:~~
	~~:<math>\tau_1 \subseteq \tau_2</math>~~.
	~~That is, every element of τ<sub>1</sub> is also an element of τ<sub>2</sub>~~. ~~Then the topology τ<sub>1</sub> is said~~ to be a '''coarser''' ('''weaker''' or '''smaller''') '''topology''' than τ<sub>2</sub>, and τ<sub>2</sub> is said to be a '''finer''' ('''stronger''' or '''larger''') '''topology''' than τ<sub>1</sub>.
	~~<ref group="nb">There are some authors, especially [[mathematical analysis\|analyst]]s, who use the terms ''weak'' and ''strong'' with opposite meaning (Munkres, p. 78).</ref>~~
	~~If additionally~~
	~~:<math>\tau_1 \neq \tau_2</math>~~
	~~we say τ<sub>1</sub> is '''strictly coarser''' than τ<sub>2</sub> and τ<sub>2</sub> is '''strictly finer''' than τ<sub>1</sub>.<ref name="Munkres"/>~~

	~~The [[binary relation]] ⊆ defines a [[partial ordering relation]] on the set~~ of ~~all possible topologies on ''X''.~~

	~~== Examples ==~~

	~~The finest topology on ''X'' is the [[discrete topology]]~~; ~~this topology makes all subsets open. The coarsest topology on ''X'' is the [[trivial topology]]; this topology only admits the null set~~
	~~and the whole space as open sets.~~

	In [[function space]]s and spaces of [[Measure (mathematics)\|measures]] there are often a number of possible topologies. See [[topologies on the set of operators on a Hilbert space]] for some intricate relationships.

	All possible [[polar topology\|polar topologies]] on a [[dual pair]] are finer than the [[weak topology (polar topology)\|weak topology]] and coarser than the [[strong topology (polar topology)\|strong topology]].

	~~== Properties ==~~

	~~Let τ<sub>1</sub> and τ<sub>2</sub> be two topologies on a set ''X''. Then the following statements are equivalent:~~
	* τ<sub>1</sub> ⊆ τ<sub>2</sub>
	* the [~~[identity function\|identity map]] id<sub>X</sub>~~ : ~~(''X'', τ<sub>2<~~/~~sub>) → (''X'', τ<sub>1<~~/~~sub>) is a [[continuous map (topology)\|continuous map]]~~.
	* the identity map id<sub>X</sub> : (''X'', τ<sub>1</sub>) → (''X'', τ<sub>2</sub>) is an [[open map]] (or, equivalently, a [[closed map]])

	~~Two immediate corollaries of this statement are~~
	*A continuous map ''f'' : ''X'' → ''Y'' remains continuous if the topology on ''Y'' becomes ''coarser'' or the topology on ''X'' ''finer''.
	*An open (resp. closed) map ''f'' : ''X'' → ''Y'' remains open (resp. closed) if the topology on ''Y'' becomes ''finer'' or the topology on ''X'' ''coarser''.

	One can also compare topologies using [[neighborhood base]]s. Let τ<sub>1</sub> and τ<sub>2</sub> be two topologies on a set ''X'' and let ''B''<sub>''i''</sub>(''x'') be a local base for the topology τ<sub>''i''</sub> at ''x'' ∈ ''X'' for ''i'' = 1,2. Then τ<sub>1</sub> ⊆ τ<sub>2</sub> if and only if for all ''x'' ∈ ''X'', each open set ''U''<sub>1</sub> in ''B''<sub>1</sub>(''x'') contains some open set ''U''<sub>2</sub> in ''B''<sub>2</sub>(''x''). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

	~~==Lattice of topologies==~~

	The set of all topologies on a set ''X'' together with the partial ordering relation ⊆ forms a [[complete lattice]] that is also closed under arbitrary intersections. That is, any collection of topologies on ''X'' have a ''meet'' (or [[infimum]]) and a ''join'' (or [[supremum]]). ~~The meet of a collection of topologies is the [[intersection (set theory)\|intersection]~~] ~~of those topologies. The join, however~~, ~~is not generally the [[union (set theory)\|union]] of those topologies (the union of two topologies need not be a topology) but rather the topology [[subbase\|generated by]] the union.~~

	Every complete lattice is also a [[bounded lattice]], which is to say that it has a [[greatest element\|greatest]] and [[least element]]. In the case of topologies, the greatest element is the [[discrete topology]] and the least element is the [[trivial topology]].

	~~== Notes ==~~
	~~{{reflist\|group=nb}}~~

	~~== See also ==~~

	* [[Initial topology]], the coarsest topology on a set to make a family of mappings from that set continuous
	* [[Final topology]], the finest topology on a set to make a family of mappings into that set continuous

	~~==References==~~
	~~{{Reflist\|refs=~~
	~~<ref name="Munkres">~~
	~~{{cite book \| last = Munkres \| first = James R. \| authorlink = James Munkres~~
	~~\| title = Topology \| edition = 2nd~~
	~~\| publisher = [[Prentice Hall]] \| location = Saddle River, NJ \| year = 2000~~
	~~\| isbn = 0-13-181629-2~~
	~~\| pages = 77–78 }}~~
	~~</ref>~~
	}}

	~~[[Category:General topology]]~~

en>Biff.Martin at 05:56, 2 December 2013

2013-12-02T05:56:47Z

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en>Cydebot: Robot - Speedily moving category Media Access Control to :Category:Media access control per CFDS.

2012-03-03T23:10:52Z

Robot - Speedily moving category Media Access Control to Category:Media access control per CFDS.

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