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9 papers
Invariant Ideal Axiom
Michael Hrušák, Alexander Shibakov
We introduce and prove the consistency of a new set theoretic axiom we call the \emph{Invariant Ideal Axiom}. The axiom enables us to provide (consistently) a full topological clas…
Precompact groups and convergence
Alexander Shibakov
We consider precompact sequential and Fréchet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question o…
Countably compact groups and sequential order
Dmitri Shakhmatov, Alexander Shibakov
We use to construct, for every a sequential countably compact topological group of sequential order . This establishes the independence of the existenc…
Scott functions, their representations on domains, and applications to random sets
Motoya Machida, Alexander Shibakov
Choquet theorems (1954) on integral representation for capacities are fundamental to probability theory. They inspired a growing body of research into different approaches and gene…
Convergence in topological groups and the Cohen reals
Alexander Shibakov
We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding Cohen re…
Stack and register complexity of radix conversions
Motoya Machida, Alexander Shibakov
We investigate the question of computational resources (such as stacks and counters) necessary to perform radix conversions. To this end it is shown that no PDA can compute the sig…
On large sequential groups
Alexander Shibakov
We construct, using , an example of a sequential group such that the only countable sequential subgroups of are closed and discrete, and the only quotients of…
On sequential analytic groups
Alexander Shibakov
We answer a question of S.~Todorčević and C.~Uzcátegui from \cite{TU1} by showing that the only possible sequential orders of sequential analytic groups are and . Other re…
No interesting sequential groups
Alexander Shibakov
We prove that it is consistent with ZFC that no sequential topological groups of intermediate sequential orders exist. This shows that the answer to a 1981 question of P.~Nyikos is…