Compact Semitopological Semigroups: An Intrinsic Theory

Ruppert, Wolfgang

doi:10.1007/bfb0073675

Cited by 138 publications

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“…We shall follow the terminology of [2,3,4,5,15]. If X is a topological space and A ⊆ X, then by cl X (A) and int X (A) we denote the topological closure and interior of A in X, respectively.…”

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confidence: 99%

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

Гутік¹,

Sobol²

2016

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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.

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confidence: 99%

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

Гутік¹,

Sobol²

2016

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“…Indeed, by [14, Lemma 1.2.6], any idempotent lies below a <K-maximal idempotent. (Be cautioned that the continuity is reversed in [14] from that which we are using.) By [13,Theorem 3.3], for any <R-maximal idempotent p, {x G ßN : p = x + p} is finite.…”

Section: Decreasing Chains Of Idempotentsmentioning

confidence: 90%

Chains of idempotents in $\beta\bold N$

Hindman¹,

Strauss²

1995

Proc. Amer. Math. Soc.

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“…An idempotent p of a semigroup S is right maximal if for every idempotent q ∈ S, p ≤ R q implies q ≤ R p. Every compact Hausdorff right topological semigroup has a right maximal idempotent [8,Theorem 2.7]. An ultrafilter u on κ is countably complete if whenever {A n : n < ω} is a partition of κ, there is n < ω such that …”

Section: Lemma 32 ([6]) Let S Be a Group And Let P ∈ S * Then Thementioning

confidence: 99%