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| In [[axiomatic set theory]], the '''Rasiowa–Sikorski lemma''' (named after [[Helena Rasiowa]] and [[Roman Sikorski]]) is one of the most fundamental facts used in the technique of [[forcing (mathematics)|forcing]]. In the area of forcing, a subset ''D'' of a forcing notion (''P'', ≤) is called '''dense in ''P''''' if for any ''p'' ∈ ''P'' there is ''d'' ∈ ''D'' with ''d'' ≤ ''p''. A [[filter (mathematics)|filter]] ''F'' in ''P'' is called ''D''-[[generic filter|generic]] if
| | Hi, everybody! My name is Felicitas. <br>It is a little about myself: I live in Australia, my city of Towaninny South. <br>It's called often Northern or cultural capital of VIC. I've married 4 years ago.<br>I have two children - a son (Lena) and the daughter (Cole). We all like Urban exploration.<br><br>Check out my web-site - [http://www.jsuttoncom.com www.jsuttoncom.com] |
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| :''F'' ∩ ''E'' ≠ ∅ for all ''E'' ∈ ''D''. | |
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| Now we can state the '''Rasiowa–Sikorski lemma''':
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| :Let (''P'', ≤) be a [[poset]] and ''p'' ∈ ''P''. If ''D'' is a [[countable]] family of [[Dense order|dense]] subsets of ''P'' then there exists a ''D''-generic [[filter (mathematics)|filter]] ''F'' in ''P'' such that ''p'' ∈ ''F''.
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| == Proof of the Rasiowa–Sikorski lemma ==
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| The proof runs as follows: since ''D'' is countable, one can enumerate the dense subsets of ''P'' as ''D''<sub>1</sub>, ''D''<sub>2</sub>, …. By assumption, there exists ''p'' ∈ ''P''. Then by density, there exists ''p''<sub>1</sub> ≤ ''p'' with ''p''<sub>1</sub> ∈ ''D''<sub>1</sub>. Repeating, one gets … ≤ ''p''<sub>2</sub> ≤ ''p''<sub>1</sub> ≤ ''p'' with ''p''<sub>''i''</sub> ∈ ''D''<sub>''i''</sub>. Then ''G'' = { ''q'' ∈ ''P'': ∃ ''i'', ''q'' ≥ ''p''<sub>''i''</sub>} is a ''D''-generic filter.
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| The Rasiowa–Sikorski lemma can be viewed as a weaker form of an equivalent to [[Martin's axiom]]. More specifically, it is equivalent to MA(<math>\aleph_0</math>).
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| == Examples ==
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| *For (''P'', ≥) = (Func(''X'', ''Y''), ⊂), the poset of [[partial function]]s from ''X'' to ''Y'', define ''D''<sub>''x''</sub> = {''s'' ∈ ''P'': ''x'' ∈ dom(''s'')}. If ''X'' is countable, the Rasiowa–Sikorski lemma yields a {''D''<sub>''x''</sub>: ''x'' ∈ ''X''}-generic filter ''F'' and thus a function ∪ ''F'': ''X'' → ''Y''.
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| *If we adhere to the notation used in dealing with ''D''-[[generic filter]]s, {''H'' ∪ ''G''<sub>0</sub>: ''P''<sub>''ij''</sub>''P''<sub>''t''</sub>} forms an ''H''-[[generic filter]].
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| *If ''D'' is uncountable, but of [[cardinality]] strictly smaller than <math>2^{\aleph_0}</math> and the poset has the [[countable chain condition]], we can instead use [[Martin's axiom]].
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| == See also ==
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| *[[Generic filter]]
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| *[[Martin's axiom]]
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| == References ==
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| * ''Set Theory for the Working Mathematician''. Ciesielski, Krzysztof. Cambridge University Press, 1997. ISBN 0-521-59465-0
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| * {{cite book|first=Kenneth|last=Kunen|authorlink=Kenneth Kunen|title=[[Set Theory: An Introduction to Independence Proofs]]|publisher=North-Holland|year=1980|isbn=0-444-85401-0}}
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| == External links ==
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| * Tim Chow's newsgroup article [http://www-math.mit.edu/~tchow/mathstuff/forcingdum Forcing for dummies] is a good introduction to the concepts and ideas behind forcing; it covers the main ideas, omitting technical details
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| {{DEFAULTSORT:Rasiowa-Sikorski lemma}}
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| [[Category:Forcing (mathematics)]]
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| [[Category:Lemmas]]
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Hi, everybody! My name is Felicitas.
It is a little about myself: I live in Australia, my city of Towaninny South.
It's called often Northern or cultural capital of VIC. I've married 4 years ago.
I have two children - a son (Lena) and the daughter (Cole). We all like Urban exploration.
Check out my web-site - www.jsuttoncom.com