On feebly compact topologies on the semilattice \(\exp_{n_{\lambda}}\)

DOI10.15330/MS.46.1.29-43zbMATH Open1377.22003arXiv1606.00395OpenAlexW2415026776MaRDI QIDQ2407959

O. Sobol, Oleg Gutik

Publication date: 6 October 2017

Published in: Matematychni Studiï (Search for Journal in Brave)

Abstract: We study feebly compact topologies $au$ on the semilattice $left(ex{p}_{n}lambda,capight)$ such that $left(ex{p}_{n}lambda,auight)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $ex{p}_{n}lambda$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $lambda$ for a $T_1$-topology $au$ on $ex{p}_{n}lambda$ the following conditions are equivalent: $(i)$ $left(ex{p}_{n}lambda,auight)$ is a compact topological semilattice; $(ii)$ $left(ex{p}_{n}lambda,auight)$ is a countably compact topological semilattice; $(iii)$ $left(ex{p}_{n}lambda,auight)$ is a feebly compact topological semilattice; $(iv)$ $left(ex{p}_{n}lambda,auight)$ is a compact semitopological semilattice; $(v)$ $left(ex{p}_{n}lambda,auight)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology $a{u}_{operatornameextsffc}^2$ such that $left(ex{p}_{2}lambda,a{u}_{operatornameextsffc}^{2}ight)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $lambda$ every $T_1$-semiregular feebly compact semitopological semilattice $ex{p}_{n}lambda$ is a compact topological semilattice.

Full work available at URL: https://arxiv.org/abs/1606.00395

zbMATH Keywords

compactcountably compacttopological semilatticefeebly compactsemiregular spacesemitopological semilattice

Mathematics Subject Classification ID

Compactness (54D30) Topological lattices, etc. (topological aspects) (54H12) Lower separation axioms ((T_0)--(T_3), etc.) (54D10) Structure of topological semigroups (22A15) Topological semilattices, lattices and applications (22A26)

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