Reverse Mathematics of Mf Spaces

Stephan

2010

Michigan Math. J.

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We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T 1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.

Section: Cardinality Of Poset Spacesmentioning

confidence: 84%

“…There are also second-countable nonmetrisable Hausdorff MF spaces. One example is the Gandy-Harrington space from modern descriptive set theory (see [11]).…”

Section: Poset Spacesmentioning

confidence: 99%

Topological aspects of Poset spaces

Stephan

2010

Michigan Math. J.

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“…Indeed, for any class Γ we have that Γ-CE implies Γ-FCP, because any instance of the latter can be regarded as an instance of the former by adding an empty finitary closure operator. Conversely, if Γ is Π 0 n , Π 1 n , Σ 1 n , or ∆ 1 n , then Γ-FCP is equivalent to Γ-CA 0 by Theorem 2.3 (2), and hence equivalent to Γ-CE. Thus, in particular, parts (2)-(5) of Corollary 2.5 hold for CE in place of FCP, and the full scheme CE itself is equivalent to Z 2 .…”

Section: Finitary Closure Operatorsmentioning

confidence: 99%

Reverse mathematics and properties of finite character

Dzhafarov

Annals of Pure and Applied Logic

2012

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We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.

“…Downey [1] gives a thorough description of many results. We note that Mummert has shown [2] that there is a computable poset with a computable filter F such that the complete Σ 1 1 set is one-one reducible to any extension of F to a maximal filter.…”

Section: Introductionmentioning

confidence: 98%

Filters on Computable Posets

Lempp

Notre Dame J. Formal Logic

2006

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We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a ∆ 0 2 maximal filter, and there is a computable poset with no Π 0 1 or Σ 0 1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle "every countable poset has a maximal filter" is equivalent to ACA 0 over RCA 0. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a Σ 0 1 unbounded filter, and there is a computable poset with no Π 0 1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA 0 over RCA 0. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets.