Search arXivSearch

arXiv · 2402.01418

Extensions and factorizations of topological and semitopological universal algebras

Abstract

The possibility of extending operations of topological and semitopological algebras to their Stone-Čech compactification and factorization of continuous functions through homomorphisms to metrizable algebras are investigated. Most attention is paid to pseudocompact and compact algebras.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Evgenii Reznichenko. 2024-06-08. Extensions and factorizations of topological and semitopological universal algebras. https://arxiv.org/abs/2402.01418

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Three Problems on Separable Quotients of Precompact Abelian Groups

We address three problems on separable quotients of topological groups posed by Leiderman, Morris, and Tkachenko in \cite{LMT} published on Israel Journal of Mathematics. First, we construct in ZFC a connected Baire Pontryagin-reflexive dense subgroup of $\T^{\cc}$ whose countable subgroups are $h$-embedded and whose uncountable subgroups are dense. Its underlying abstract group is the circle group, and all its compact subsets are finite. Second, we construct a zero-dimensional Baire Pontryagin-reflexive example with the same subgroup properties whose underlying group is free abelian of rank $\cc$. Both examples have no nontrivial separable Hausdorff quotient. Third, for the group constructed in their Theorem~3.5, we determine every closed subgroup of every finite power up to an integral change of coordinates and prove that every countable subgroup of every Hausdorff quotient of a finite power is $h$-embedded and closed. The same conclusions hold for our free Baire reflexive example. These results answer Problem~1.25 negatively, Problem~3.12 affirmatively and realize all three regularity properties in Problem~3.14 simultaneously in \cite{LMT}.

math.GN

Topological Vector Group Topologies Between the Minimal Topology and the Usual Topology on the Real Line

For every positive sequence that tends to zero faster than every fixed exponential, we construct a Hausdorff topological Vector Group topology on the additive group of real numbers. It lies strictly between the minimal Hausdorff topological Vector Group topology and the usual topology. The construction is illustrated by factorial powers, quadratic exponential decay, and prime radicals divided by a quadratic exponential. We also record a finite scalar covering criterion for comparing two such topologies.

math.GN

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN