Papkovich–Neuber solution: Difference between revisions
Jump to navigation
Jump to search
en>TomyDuby m →References: Fixed a typo |
en>Yobot m WP:CHECKWIKI error fixes - Replaced endash with hyphen in sortkey per WP:MCSTJR using AWB (9100) |
||
| Line 1: | Line 1: | ||
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}} | |||
Latest revision as of 17:52, 20 April 2013
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 and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922.[1] The problem gained wide exposure three decades later as an exercise in John L. Kelley's classic textbook General Topology.[2]
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 idempotent.)
(2) ccS = S. (The complement operation is an 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 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:
![{\displaystyle (0,1)\cup (1,2)\cup \{3\}\cup {\bigl (}[4,5]\cap \mathbb {Q} {\bigr )},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f7724035dc07d8e83b1e51da4248750f7ad4a6)
where denotes an open interval and denotes a closed interval.
![{\displaystyle [4,5]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29536dee83347f147536b73557908dfe3761c34d)
Further results
Despite its origin within the context of a topological space, Kuratowski's closure-complement problem is actually more algebraic 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.[3]
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
- The Kuratowski Closure-Complement Theorem by B. J. Gardner and Marcel Jackson
- The Kuratowski Closure-Complement Problem by Mark Bowron
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang