Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComputable Stone spaces
167
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 covering dimension extends to higher dimensions and to closed shrinkings of finite open covers. To deal with zero-dimensional subspaces uniformly, four operations (relative to the space and a class of subspaces) are defined; these correspond to definitions of inductive and covering dimensions and a countable basis condition. Finally, an effective retract characterisation of zero-dimensionality is proven under an effective compactness condition. In one direction this uses a version of the construction of bilocated sets.
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 Stone quotients, closed subalgebras and finite direct products. Looking at Stone spaces as the dual spaces of Boolean algebras, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which, in the case of a Stone signature, culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for residually closed Stone varieties is also established.
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 of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $omega_2$ countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentble by a forcing notion that does not make $omega_2$ countable has a copy in the ground model. We also show that any countable structure $mathcal{A}$ that is generically presentable by a forcing notion not collapsing $omega_1$ has a countable copy in $V$, as does any structure $mathcal{B}$ generically Muchnik reducible to a structure $mathcal{A}$ of cardinality $aleph_1$. The former positive result yields a new proof of Harringtons result that counterexamples to Vaughts conjecture have models of power $aleph_1$ with Scott rank arbitrarily high below $omega_2$. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.
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, the class of coherent completions of Peano Arithmetic or the class of reals of minimal degrees. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kucera to be non-negligible in a strong sense: every Martin-Lof random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Lof random real. In this paper, we show that such the set of such DNC functions is in fact non-negligible. More precisely, we prove that for every sufficiently fast-growing computable~$h$, every 2-random real computes an $h$-bounded DNC function which computes no Martin-Lof random real. Further, we show that the same holds for the set of reals which compute a DNC function but no bounded DNC function. The proofs of these results use a combination of a technique due to Kautz (which, following a metaphor of Shen, we like to call a `fireworks argument) and bushy tree forcing, which is the canonical forcing notion used in the study of DNC functions.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
