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On the Lattices of Effectively Open Sets

Oleg Kudinov and Victor Selivanov CCC-Workshop, Kochel, September 2015

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

The lattice E of c.e. sets is a popular object of study in Computability Theory [So87]. A principal fact about this lattice is the undecidability of its first-order theory Th(E). Moreover, Th(E) is known [HN98] to be m-equivalent to the first-order arithmetic Th(N) where N := (ω; +, ×). The lattice E may be considered as the lattice of effectively open subsets of the discrete topological space ω. The lattices of effectively open sets Σ0

1(X) of various “effective” topological

spaces X are of interest in Computable Analysis [We00] and Effective Descriptive Set Theory, hence it is natural to ask which results about E hold true for the lattices of effectively open sets of “natural” effective topological spaces. Here we attempt to make apparently first steps in this direction.

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

First we show that for many natural effective spaces X the theory Th(Σ0

1(X)) is undecidable. By strongly computable ϕ-space

(SCPS) we mean an effective (not necessarily complete) ϕ-space [Se06] X such that the specialization order is computable on the compact elements, and there is a computable infinite sequence of pairwise incomparable compact elements. Many popular domains are SCPSs. In particular, Pω and the domains of partial continuous functionals of finite types are SCPSs. Theorem Let X be a SCPS or a non-singleton computable metric space [We00] without isolated points. Then Th(Σ0

1(X)) is hereditarily

undecidable.

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

Next we discuss the complexity of Th(Σ0

1(X)). First we establish a

natural upper bound that applies to many locally compact spaces. By arithmetically locally compact space (ALCS) we mean a triple (X, β, κ) consisting of a countably based space X, a numbering β

• f a base in X, and a numbering κ of compact sets in X such that

any set βn is a union of some sets in {κi | i < ω}, and the relation “κi ⊆ βn0 ∪ · · · ∪ βnk” is arithmetical. Many popular locally compact spaces are ALCSs. In particular, the effective ϕ-spaces and the finite dimensional Euclidean spaces are ALCSs. Proposition If X is an ALCS then Th(Σ0

1(X)) ≤m Th(N).

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

The precise estimation of the complexity of Th(Σ0

1(X)) is very

subtle and strongly depends on the topology of X. So far we were able to precisely characterize the complexity of Th(Σ0

1(X)) only for

small classes of spaces, e.g. we have the following result: Theorem Let X be the space of reals R, or the domain Pω, or the domain of partial continuous functionals of a given finite type. Then Th(Σ0

1(X)) ≡m Th(N).

For many natural spaces X we still have a big gap between the known (to us) lower and the upper bound for Th(Σ0

1(X)). In

particular, this is the case even for the Baire space.

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

Fortunately, for Cantor space C there is a precise description of the theory T = Th(Σ0

1(C)) as a consequence of Stone duality and one

• f deep result by A. Nies. In according to Stone duality, open

subsets of 0-dimensional compact T2−space X bijectively correspond to ideals of the boolean algebra B(X) of clopen subsets

• f X. If additionally X is computable metric space and B(X) is

computable, then effectively open subsets of X correspond to c.e. ideals of B(X) and computable open subsets (as complements to computable closed ones) correspond to computable ideals! Since B(C) is just atomless boolean algebra Bη, the structure Σ0

1(X) is isomorphic to the structure of c.e. ideals of Bη. It is

shown in [N00] that the first-order arithmetic is interpretable in this structure, so, Th(N) ≤m T. And, since both structures are arithmetical, these arguments lead to the following Proposition Th(Σ0

1(C)) ≡m Th(N).

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets

• L. Harrington and A. Nies. Coding in the lattice of enumerable
• sets. Advances in Mathematics, 133 (1998), 133-162.
• A. Nies. Effectively dense boolean algebras and their applications.

Transactions AMS, 352 (2000), N 11, 4989-5012. V.L. Selivanov. Towards a descriptive set theory for domain-like

• structures. Theoretical Computer Science, 365 (2006), 258–282.

R.I. Soare. Recursively Enumerable Sets and Degrees. Berlin, Springer, 1987.

• K. Weihrauch. Computable Analysis. Berlin, Springer, 2000.

Oleg Kudinov and Victor Selivanov On the Lattices of Effectively Open Sets