Computably regular topological spaces

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces

Computable Separation in Topology, from T_0 to T_3

This article continues the study of computable elementary topology started in (Weihrauch, Grubba 2009). We introduce a number of computable versions of the topological $T_0$ to $T_3$ separation axioms and solve their logical relation completely. In particular, it turns out that computable $T_1$ is equivalent to computable $T_2$. The strongest axiom $SCT_3$ is used in (Grubba, Schroeder, Weihrauch 2007) to construct a computable metric

First Order Theories of Some Lattices of Open Sets

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic

Characterization theorem for the conditionally computable real functions

The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable real functions with respect to the same class of operators is a proper extension of the class of uniformly computable real functions and it computes the elementary functions of calculus on their whole domains. The definition of both classes relies on certain transformations of infinitistic names of real numbers. In the present paper, the conditional computability of real functions is characterized in the spirit of Tent and Ziegler, avoiding the use of infinitistic names

Computability of probability measures and Martin-Lof randomness over metric spaces

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).Comment: 29 page

Kleinian groups and the rank problem

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper12.abs.htm