A note on universal minimal dynamical systems
Let denote the phase space of the universal minimal dynamical system for a group . Our aim is to show that is homeomorphic to the absolute of , whenever is a countable Abelian group.
Universal minimal dynamical system for reals
Our aim is to give a description of and , the phase space of universal ambit and the phase space of universal minimal dynamical system for the group of real numbers with the usual topology.
Minimal dynamical systems for -bounded groups
A decomposition theorem for compact groups with an application to supercompactness
We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.
Boolean algebras admitting a countable minimally acting group
The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra.
Characterizing chainable, tree-like, and circle-like continua
We prove that a continuum X is tree-like (resp. circle-like, chainable) if and only if for each open cover 𝓤₄ = {U₁,U₂,U₃,U₄} of X there is a 𝓤₄-map f: X → Y onto a tree (resp. onto the circle, onto the interval). A continuum X is an acyclic curve if and only if for each open cover 𝓤₃ = {U₁,U₂,U₃} of X there is a 𝓤₃-map f: X → Y onto a tree (or the interval [0,1]).
Topological multidimensional van der Waerden theorem
On continuous self-maps and homeomorphisms of the Golomb space
The Golomb space is the set of positive integers endowed with the topology generated by the base consisting of arithmetic progressions with coprime . We prove that the Golomb space has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set of prime numbers is a dense metrizable subspace of , and each homeomorphism of has the following properties: , , , and for all . Here and denotes the set of prime divisors...
The Golomb space is topologically rigid
The Golomb space is the set of positive integers endowed with the topology generated by the base consisting of arithmetic progressions with coprime . We prove that the Golomb space is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by T. Banakh at Mathoverflow in 2017.
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