en>Yobot: WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

2013-04-20T15:52:27Z

WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

← Older revision		Revision as of 17:52, 20 April 2013
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	~~Oscar is how he~~'s ~~known as~~ and ~~he completely enjoys this name~~. ~~Years~~ in the ~~past we moved to Puerto Rico~~ and ~~my family loves it~~. ~~Managing individuals~~ is ~~his profession~~. He is ~~really fond~~ of ~~doing ceramics but he~~ is ~~having difficulties~~ to ~~find time for it~~.<br><br>~~Feel free~~ to ~~surf to my web blog;~~ [http://~~www~~.~~hotporn123~~.~~com~~/~~blog~~/~~154316 std testing at home~~]		In [[point-set topology]], '''Kuratowski's closure-complement problem''' asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of [[closure (topology)\|closure]] and [[complement (set theory)\|complement]] to a given starting subset of a [[topological space]]. The answer is 14. This result was first published by [[Kazimierz Kuratowski]] in 1922.<ref>{{Cite journal
			\| last = Kuratowski
			\| first = Kazimierz
			\| authorlink = Kazimierz Kuratowski
			\| title = Sur l'operation A de l'Analysis Situs
			\| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf
			\| journal = Fundamenta Mathematicae
			\| volume = 3
			\| pages = 182–199
			\| publisher = Polish Academy of Sciences
			\| location = Warsaw
			\| year = 1922
			\| issn = 0016-2736}}</ref> The problem gained wide exposure three decades later as an exercise in [[John L. Kelley]]'s classic textbook ''General Topology''.<ref>{{Cite book
			\| last = Kelley
			\| first = John
			\| isbn = 0-387-90125-6
			\| authorlink = John L. Kelley
			\| title = General Topology
			\| publisher = Van Nostrand
			\| year = 1955
			\| page = 57}}</ref>

			==Proof==

			Letting ''S'' denote an arbitrary subset of a topological space, write ''kS'' for the closure of ''S'', and ''cS'' for the complement of ''S''. The following three identities imply that no more than 14 distinct sets are obtainable:

			(1) ''kkS'' = ''kS''. (The closure operation is [[idempotency\|idempotent]].)

			(2) ''ccS'' = ''S''. (The complement operation is an [[involution (mathematics)\|involution]].)

			(3) ''kckckckcS'' = ''kckcS''.(Or equivalently ''kckckckS=kckckckccS=kckS''. Using identity (2).)

			The first two are trivial. The third follows from the identity ''kikiS'' = ''kiS'' where ''iS'' is the [[interior (topology)\|interior]] of ''S'' which is equal to the complement of the closure of the complement of ''S'', ''iS'' = ''ckcS''. (The operation ''ki'' = ''kckc'' is idempotent.)

			A subset realizing the maximum of 14 is called a '''14-set'''. The space of [[real numbers]] under the usual topology contains 14-sets. Here is one example:
			:<math>(0,1)\cup(1,2)\cup\{3\}\cup\bigl([4,5]\cap\Q\bigr),</math>
			where <math>(1,2)</math> denotes an [[Interval_(mathematics)#Terminology\|open interval]] and <math>[4,5]</math> denotes a closed interval.

			==Further results==

			Despite its origin within the context of a topological space, Kuratowski's closure-complement problem is actually more [[algebra]]ic than topological. A surprising abundance of closely related problems and results have appeared since 1960, many of which have little or nothing to do with point-set topology.<ref>{{Cite journal
			\| last = Hammer
			\| first = P. C.
			\| title = Kuratowski's Closure Theorem
			\| journal = Nieuw Archief voor Wiskunde
			\| volume = 8
			\| publisher = Royal Dutch Mathematical Society
			\| pages = 74–80
			\| year = 1960
			\| issn = 0028-9825}}</ref>

			==References==
			{{Reflist}}

			== External links ==
			* [http://nzjm.math.auckland.ac.nz/images/6/63/The_Kuratowski_Closure-Complement_Theorem.pdf The Kuratowski Closure-Complement Theorem] by B. J. Gardner and Marcel Jackson

			* [http://mathdl.maa.org/mathDL/60/?pa=content&sa=viewDocument&nodeId=3343&pf=1 The Kuratowski Closure-Complement Problem] by Mark Bowron

			[[Category:Topology]]
			[[Category:Mathematical problems]]

			{{topology-stub}}

en>TomyDuby: /* References */ Fixed a typo

2012-03-27T20:50:26Z

References: Fixed a typo

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Oscar is how he's known as and he completely enjoys this name. Years in the past we moved to Puerto Rico and my family loves it. Managing individuals is his profession. He is really fond of doing ceramics but he is having difficulties to find time for it.<br><br>Feel free to surf to my web blog; [http://www.hotporn123.com/blog/154316 std testing at home]

Papkovich–Neuber solution - Revision history

en>Yobot: WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100)

en>TomyDuby: /* References */ Fixed a typo