ﻻ يوجد ملخص باللغة العربية
We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space $X$, the Banach space $C(X;mathbb{R})$ has a computable presentation if, and only if, $X$ is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.
We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that concerns c
Profinite algebras are the residually finite compact algebras, whose underlying topological spaces are Stone spaces. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed signature that are closed under taking
In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {em generic Muchnik reducibility} that can be used to to compare the complexity o
A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, t
We obtain a computable structure of Scott rank omega_1^{CK} (call this ock), and give a general coding procedure that transforms any hyperarithmetical structure A into a computable structure A such that the rank of A is ock, ock+1, or < ock iff the same is true of A.