Field desorption - Revision history

en>Monkbot: /* Further reading */Task 6f: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

2014-11-18T00:22:43Z

Further reading: Task 6f: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

← Older revision		Revision as of 02:22, 18 November 2014
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	~~In [[mathematics]], a [[subset]] <math>A</math> of a [[topological space]] is said to be '''dense-in-itself''~~' ~~if <math>A</math> contains no [[isolated point]]~~s.		The author's title is Christy Brookins. For years he's been residing in Alaska and he doesn't strategy on altering it. My working day occupation is a travel agent. To play lacross is the factor I adore most of all.<br><br>My weblog - real psychics ([http://kard.dk/?p=24252 kard.dk])

	~~Every dense-in-itself [[closed set]]~~ is ~~[[perfect set\|perfect]]~~. ~~Conversely, every perfect set is dense-~~in~~-itself~~.

	~~A simple example of a set which~~ is ~~dense-in-itself but not closed (and hence not~~ a ~~perfect set) is the subset of [[irrational numbers]] (considered as a subset of the [[real numbers]])~~. ~~This set~~ is ~~dense-in-itself because every [[Neighbourhood (mathematics)\|neighborhood]] of an irrational number <math>x</math> contains at least one other irrational number <math>y \neq x</math>. On~~ the ~~other hand, this set~~ of ~~irrationals is not closed because every [[rational number]] lies in its [[closure (topology)\|closure]]~~. ~~For similar reasons, the set of rational numbers (also considered as a subset of the [[real numbers]]) is also dense-in-itself but not closed.~~

	~~The above examples, the irrationals and the rationals, are also [[dense set]]s in their topological space, namely~~ <~~math~~>~~\mathbb{R}~~<~~/math~~>~~. As an example that is dense~~-~~in-itself but not dense in its topological space, consider <math>\mathbb{Q} \cap~~ [~~0,1]<~~/~~math>. This set is not dense in <math>\mathbb{R}<~~/~~math> but is dense-in-itself~~.

	~~It is also interesting to note, although tautological, that the domain of a [[continuous function]] must be the union of dense-in-itself sets and~~/~~or isolated points.{{fact\|date=January 2013}}~~

	~~==See also==~~
	* [[Nowhere dense set]]
	* [[Dense order]]
	* [[Perfect space]]
	* [[Glossary of topology]]

	~~==References==~~
	~~{{reflist}}~~
	*{{cite book \| 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 \| page=6 }}~~

	~~{{PlanetMath attribution\|id=6228\|title=Dense in-itself}}~~

	~~[[Category:Topology]]~~

	~~{{topology-stub}}~~

en>Kahleflower93: added ionization link, wrote FI was soft ionization source

2013-11-17T21:20:02Z

added ionization link, wrote FI was soft ionization source

← Older revision		Revision as of 23:20, 17 November 2013
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	~~The name~~ of the ~~author~~ is ~~Jayson~~. ~~My day occupation~~ is ~~an invoicing officer~~ but ~~I've already applied for another 1~~. ~~My wife~~ and ~~I live~~ in ~~Mississippi but now I'm considering other options~~. ~~What me and my family adore~~ is ~~performing ballet~~ but ~~I've been taking on new issues lately~~.<br><br>~~Feel free~~ to ~~visit my site~~ [~~http:~~/~~/203~~.~~250~~.78.~~160/zbxe/?document_srl~~=~~1792908 psychic chat online~~]		In [[mathematics]], a [[subset]] <math>A</math> of a [[topological space]] is said to be '''dense-in-itself''' if <math>A</math> contains no [[isolated point]]s.

			Every dense-in-itself [[closed set]] is [[perfect set\|perfect]]. Conversely, every perfect set is dense-in-itself.

			A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of [[irrational numbers]] (considered as a subset of the [[real numbers]]). This set is dense-in-itself because every [[Neighbourhood (mathematics)\|neighborhood]] of an irrational number <math>x</math> contains at least one other irrational number <math>y \neq x</math>. On the other hand, this set of irrationals is not closed because every [[rational number]] lies in its [[closure (topology)\|closure]]. For similar reasons, the set of rational numbers (also considered as a subset of the [[real numbers]]) is also dense-in-itself but not closed.

			The above examples, the irrationals and the rationals, are also [[dense set]]s in their topological space, namely <math>\mathbb{R}</math>. As an example that is dense-in-itself but not dense in its topological space, consider <math>\mathbb{Q} \cap [0,1]</math>. This set is not dense in <math>\mathbb{R}</math> but is dense-in-itself.

			It is also interesting to note, although tautological, that the domain of a [[continuous function]] must be the union of dense-in-itself sets and/or isolated points.{{fact\|date=January 2013}}

			==See also==
			* [[Nowhere dense set]]
			* [[Dense order]]
			* [[Perfect space]]
			* [[Glossary of topology]]

			==References==
			{{reflist}}
			*{{cite book \| 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 \| page=6 }}

			{{PlanetMath attribution\|id=6228\|title=Dense in-itself}}

			[[Category:Topology]]

			{{topology-stub}}

en>Yobot: WP:CHECKWIKI error fixes (category with space) + general fixes using AWB (7796)

2011-07-29T10:43:15Z

WP:CHECKWIKI error fixes (category with space) + general fixes using AWB (7796)

New page

The name of the author is Jayson. My day occupation is an invoicing officer but I've already applied for another 1. My wife and I live in Mississippi but now I'm considering other options. What me and my family adore is performing ballet but I've been taking on new issues lately.<br><br>Feel free to visit my site [http://203.250.78.160/zbxe/?document_srl=1792908 psychic chat online]