On feebly compact topologies on the semilattice $\exp_n\lambda$

Abstract

We study feebly compact topologies τ on the semilattice (exp n λ, ∩) such that (exp n λ, τ ) is a semitopological semilattice and prove that for any shift-continuous T 1 -topology τ on exp n λ the following conditions are equivalent: (i) τ is countably pracompact; Dedicated to the memory of Professor Vitaly SushchanskyyWe shall follow the terminology of [6,8,9,13]. If X is a topological space and A ⊆ X, then by cl X (A) and int X (A) we denote the closure and the interior of A in X, respectively. By ω we denote the first infinite cardinal and by N the set of positive integers.A subset A of a topological spaceWe recall that a topological space X is said to bea base consisting of regular open subsets; • compact if each open cover of X has a finite subcover; • countably compact if each open countable cover of X has a finite subcover; • countably compact at a subset A ⊆ X if every infinite subset B ⊆ A has an accumulation point x in X; • countably pracompact if there exists a dense subset A in X such that X is countably compact at A; • feebly compact (or lightly compact) if each locally finite open cover of X is finite [3]; • d-feebly compact (or DFCC ) if every discrete family of open subsets in X is finite (see [12]);• pseudocompact if X is Tychonoff and each continuous real-valued function on X is bounded. According to Theorem 3.10.22 of [8], a Tychonoff topological space X is feebly compact if and only if X is pseudocompact. Also, a Hausdorff topological space X is feebly compact if and only if every locally finite family of non-empty open subsets of X is finite [3]. Every compact space and every sequentially compact space are countably compact, every countably compact space is countably pracompact, and every countably pracompact space is feebly compact (see [2]), and every H-closed space is feebly compact too (see [10]). Also, it is obvious that every feebly compact space is d-feebly compact.A semilattice is a commutative semigroup of idempotents. On a semilattice S there exists a natural partial order: e f if and only if ef = f e = e. For any element e of a semilattice S we put ↑e = {f ∈ S : e f } .A topological (semitopological) semilattice is a topological space together with a continuous (separately continuous) semilattice operation. If S is a semilattice and τ is a topology on S such that (S, τ ) is a topological semilattice, then we shall call τ a semilattice topology on S, and if τ is a topology on S such that (S, τ ) is a semitopological semilattice, then we shall call τ a shift-continuous topology on S.

A note on feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

“…Also in [12] we proved that for an arbitrary positive integer n and an arbitrary infinite cardinal λ every shift-continuous semiregular feebly compact T 1 -topology τ on I n λ is compact. Similar results were obtained for a semitopological semilattice (exp n λ, ∩) in [23][24][25]. Also, in [26,30] it is proved that feeble compactness implies compactness for semitopological bicyclic extensions.…”

On old and new classes of feebly compact spaces

“…Our main motivation to introduce the above spaces is possible applications in topological algebra. In particular, we are going to use them in the paper [15]. 1 Selectively sequentially feebly compact Tychonoff spaces were recently introduced and studied by Dorantes-Aldama and Shakhmatov in [8].…”