Teleparallelism - Revision history

en>Cuzkatzimhut: /* Remarks */

2014-12-21T00:30:58Z

Remarks

← Older revision		Revision as of 02:30, 21 December 2014
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en>Wavelength: inserting 1 hyphen: —> "four-dimensional"—wikt:four-dimensional

2014-03-04T00:20:59Z

inserting 1 hyphen: —> "four-dimensional"—wikt:four-dimensional

← Older revision		Revision as of 02:20, 4 March 2014
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	~~In [[topology]], a branch of [[mathematics]], a '''first~~-~~countable space''' is~~ a [~~[topological space~~]~~] satisfying the "first [[axiom~~ of ~~countability]]". Specifically~~, ~~a space ''X'' is said~~ to ~~be first-countable if each point has a [[countable]] [[neighbourhood system\|neighbourhood basis]] (local base). That is~~, ~~for each point ''x'' in ''X'' there exists a [[sequence]] ''N''~~<~~sub~~>1</~~sub>, ''N''<sub>2<~~/~~sub>, … of [[neighbourhood (topology)\|neighbourhoods~~]~~] of ''x'' such~~ that ~~for any neighbourhood ''N'~~' of ~~''x'' there exists an integer ''i'' with ''N''<sub>''i''<~~/~~sub> [[subset\|contained in~~]~~] ''N''~~.		I woke up yesterday and noticed - I have been solitary for some time now and after much intimidation from friends I now locate myself registered for online dating. They promised me that there are a lot where to buy luke bryan tickets ([http://lukebryantickets.asiapak.net http://lukebryantickets.asiapak.net]) of normal, pleasant and interesting people to meet up, so here goes the pitch!<br>I strive to keep as physically fit as potential coming to the fitness center [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=several-times several-times] weekly. I appreciate my sports and strive to play or see since many a potential. I am going to frequently at Hawthorn fits being wintertime. Notice: Supposing that you would considered buying an athletics I really don't mind, I have noticed the carnage of fumbling fits at [http://www.cinemaudiosociety.org luke Bryan 2014 tickets] stocktake sales.<br>My family and buddies are awesome and spending some time with them at tavern gigabytes or dishes is consistently a must. As I find that you [http://www.ffpjp24.org How To Meet Luke Bryan] can not have a nice conversation with the [http://Photobucket.com/images/noise+I noise I] have never been into clubs. Additionally, I have two definitely cheeky and quite adorable dogs that are invariably enthusiastic to meet fresh individuals.<br><br>
	~~Since every neighborhood of any point~~ is ~~contained in an open neighborhood of~~ that ~~point the~~ [~~[neighbourhood system\|neighbourhood basis]~~] can ~~be chosen w~~.~~l.o.g~~. to ~~consist of open neighborhoods~~.

	~~==Examples and counterexamples==~~		Look into my blog post [http://lukebryantickets.omarfoundation.org luke bryan sold out]
	~~The majority of 'everyday' spaces in~~ [[mathematics]] are first-countable. In particular, every [[metric space]] is first-countable. To see this, note that the set of [[open ball]]s centered at ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''.

	~~An example of a space which is not first-countable is the [[cofinite topology]] on an uncountable set (such as the [[real line]]).~~

	Another counterexample is the [[ordinal space]] ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] where ω<sub>1</sub> is the [[first uncountable ordinal]] number. The element ω<sub>1</sub> is a [[limit point]] of the subset <nowiki>[</nowiki>0,ω<sub>1</sub>) even though no sequence of elements in <nowiki>[</nowiki>0,ω<sub>1</sub>) has the element ω<sub>1</sub> as its limit. In particular, the point ω<sub>1</sub> in the space ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] does not have a countable local base. Since ω<sub>1</sub> is the only such point, however, the subspace ω<sub>1</sub> = <nowiki>[</nowiki>0,ω<sub>1</sub>) is first-countable.

	The [[quotient space]] <math>\mathbb{R}/\mathbb{N}</math> where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a [[Sequential space#Fréchet–Urysohn space\|Fréchet-Urysohn space]].

	First-countability is strictly weaker than [[second-countability]]. Every [[second-countable space]] is first-countable, but any uncountable [[discrete space]] is first-countable but not second-countable.

	~~==Properties==~~
	One of the most important properties of first-countable spaces is that given a subset ''A'', a point ''x'' lies in the [[closure (topology)\|closure]] of ''A'' if and only if there exists a [[sequence]] {''x''<sub>''n''</sub>} in ''A'' which [[limit of a sequence\|converges]] to ''x''. This has consequences for [[limit of a function\|limits]] and [[continuity (topology)\|continuity]]. In particular, if ''f'' is a function on a first-countable space, then ''f'' has a limit ''L'' at the point ''x'' if and only if for every sequence ''x''<sub>''n''</sub> → ''x'', where ''x''<sub>''n''</sub> ≠ ''x'' for all ''n'', we have ''f''(''x''<sub>''n''</sub>) → ''L''. Also, if ''f'' is a function on a first-countable space, then ''f'' is continuous if and only if whenever ''x''<sub>''n''</sub> → ''x'', then ''f''(''x''<sub>''n''</sub>) → ''f''(''x'').

	In first-countable spaces, [[sequentially compact space\|sequential compactness]] and [[countably compact space\|countable compactness]] are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the [[Order topology\|ordinal space]] <nowiki>[</~~nowiki>0,ω<sub>1<~~/~~sub>). Every first-countable space is [[compactly generated space\|compactly generated]].~~

	~~Every [[subspace (topology)\|subspace]] of a first-countable space is first-countable~~. ~~Any countable [[product space\|product]] of a first-countable space is first-countable, although uncountable products need not be~~.

	~~==See also==~~
	*[[Second-countable space]]
	*[[Separable space]]

	~~==References==~~
	*{{Springer\|id=f/f040430\|title=first axiom of countability}}

	~~{{DEFAULTSORT:First-Countable Space}}~~
	~~[[Category:General topology]]~~
	~~[[Category:Properties of topological spaces]~~]

en>Monkbot: /* New translation teleparallel gauge theory of gravity */Fix CS1 deprecated date parameter errors

2014-01-30T16:20:56Z

New translation teleparallel gauge theory of gravity: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 18:20, 30 January 2014
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	~~Wilber Berryhill~~ is the ~~name his mothers~~ and ~~fathers gave him and he totally digs~~ that ~~name~~. ~~Alaska~~ is ~~exactly~~ where ~~he's usually been living~~. ~~Office supervising~~ is ~~my occupation~~. As a ~~woman what she really likes~~ is ~~fashion~~ and ~~she~~'~~s been doing it~~ for ~~quite~~ a ~~whilst~~.<br><br>~~Visit my web~~-~~site cheap psychic readings~~ ([~~http:~~//~~www~~.~~route41~~.~~com/excellent~~-~~advice~~-~~for~~-~~choosing~~-~~the~~-~~ideal~~-~~hobby/ website link~~])		In [[topology]], a branch of [[mathematics]], a '''first-countable space''' is a [[topological space]] satisfying the "first [[axiom of countability]]". Specifically, a space ''X'' is said to be first-countable if each point has a [[countable]] [[neighbourhood system\|neighbourhood basis]] (local base). That is, for each point ''x'' in ''X'' there exists a [[sequence]] ''N''<sub>1</sub>, ''N''<sub>2</sub>, … of [[neighbourhood (topology)\|neighbourhoods]] of ''x'' such that for any neighbourhood ''N'' of ''x'' there exists an integer ''i'' with ''N''<sub>''i''</sub> [[subset\|contained in]] ''N''.
			Since every neighborhood of any point is contained in an open neighborhood of that point the [[neighbourhood system\|neighbourhood basis]] can be chosen w.l.o.g. to consist of open neighborhoods.

			==Examples and counterexamples==
			The majority of 'everyday' spaces in [[mathematics]] are first-countable. In particular, every [[metric space]] is first-countable. To see this, note that the set of [[open ball]]s centered at ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''.

			An example of a space which is not first-countable is the [[cofinite topology]] on an uncountable set (such as the [[real line]]).

			Another counterexample is the [[ordinal space]] ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] where ω<sub>1</sub> is the [[first uncountable ordinal]] number. The element ω<sub>1</sub> is a [[limit point]] of the subset <nowiki>[</nowiki>0,ω<sub>1</sub>) even though no sequence of elements in <nowiki>[</nowiki>0,ω<sub>1</sub>) has the element ω<sub>1</sub> as its limit. In particular, the point ω<sub>1</sub> in the space ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] does not have a countable local base. Since ω<sub>1</sub> is the only such point, however, the subspace ω<sub>1</sub> = <nowiki>[</nowiki>0,ω<sub>1</sub>) is first-countable.

			The [[quotient space]] <math>\mathbb{R}/\mathbb{N}</math> where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a [[Sequential space#Fréchet–Urysohn space\|Fréchet-Urysohn space]].

			First-countability is strictly weaker than [[second-countability]]. Every [[second-countable space]] is first-countable, but any uncountable [[discrete space]] is first-countable but not second-countable.

			==Properties==
			One of the most important properties of first-countable spaces is that given a subset ''A'', a point ''x'' lies in the [[closure (topology)\|closure]] of ''A'' if and only if there exists a [[sequence]] {''x''<sub>''n''</sub>} in ''A'' which [[limit of a sequence\|converges]] to ''x''. This has consequences for [[limit of a function\|limits]] and [[continuity (topology)\|continuity]]. In particular, if ''f'' is a function on a first-countable space, then ''f'' has a limit ''L'' at the point ''x'' if and only if for every sequence ''x''<sub>''n''</sub> → ''x'', where ''x''<sub>''n''</sub> ≠ ''x'' for all ''n'', we have ''f''(''x''<sub>''n''</sub>) → ''L''. Also, if ''f'' is a function on a first-countable space, then ''f'' is continuous if and only if whenever ''x''<sub>''n''</sub> → ''x'', then ''f''(''x''<sub>''n''</sub>) → ''f''(''x'').

			In first-countable spaces, [[sequentially compact space\|sequential compactness]] and [[countably compact space\|countable compactness]] are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the [[Order topology\|ordinal space]] <nowiki>[</nowiki>0,ω<sub>1</sub>). Every first-countable space is [[compactly generated space\|compactly generated]].

			Every [[subspace (topology)\|subspace]] of a first-countable space is first-countable. Any countable [[product space\|product]] of a first-countable space is first-countable, although uncountable products need not be.

			==See also==
			*[[Second-countable space]]
			*[[Separable space]]

			==References==
			*{{Springer\|id=f/f040430\|title=first axiom of countability}}

			{{DEFAULTSORT:First-Countable Space}}
			[[Category:General topology]]
			[[Category:Properties of topological spaces]]

en>Patsobest: /* New teleparallel gravity theory */ fixed wording and made link

2012-08-30T21:25:31Z

New teleparallel gravity theory: fixed wording and made link

New page

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