On feebly compact inverse primitive (semi)topological semigroups

Гутік, Олег; Ravsky, Oleksandr

doi:10.15330/ms.44.1.3-26

Cited by 3 publications

(5 citation statements)

References 20 publications

(34 reference statements)

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“…Ravsky in [22] generalized the Comfort-Ross Theorem and proved that a Tychonoff product of an arbitrary non-empty family of feebly compact paratopological groups is feebly compact. Also, a counterpart of the Comfort-Ross Theorem for pseudocompact primitive topological inverse semigroups and primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups were proved in [11] and [14], respectively.…”

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confidence: 98%

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: 98%

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

Гутік¹,

Sobol²

2016

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“…The Comfort-Ross Theorem is generalized in [2] and it is proved that a Tychonoff product of an arbitrary non-empty family of feebly compact paratopological groups is feebly compact. Also, a counterpart of the Comfort-Ross Theorem for pseudocompact primitive topological inverse semigroups and primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups were proved in [16] and [18], respectively.…”

Section: Lemma 3 ( [12 Lemma 3])mentioning

confidence: 98%

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

Гутік¹

2022

Preprint

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“…Also there we described structure of pseudocompact topological Brandt λ 0 -extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In [20] we shown that the inversion in a quasi-regular primitive inverse pseudocompact topological semigroup is continuous. Also there, an analogue of Comfort-Ross Theorem is proved for such semigroups: the Tychonoff product of an arbitrary non-empty family of primitive inverse semiregular pseudocompact semitopological semigroups with closed maximal subgroups is a pseudocompact space, and we described the structure of the Stone-Čech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup S such that every maximal subgroup of S is a topological group.…”

Section: Introductionmentioning

confidence: 99%

“…In the paper [20] we studied the structure of inverse primitive pseudocompact semitopological and topological semigroups. We find conditions when a maximal subgroup of an inverse primitive pseudocompact semitopological semigroup S is a closed subset of S and described the topological structure of such semiregular semigroup.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2.11 implies that the map g i is continuous for every i ∈ I . Since the space i∈I A (λ i ×λ i )×S i is homeomorphic to i∈I A (λ i ×λ i )× i∈I S i , Theorem 3.2.4 from [11] and Corollary 3.3 [20] imply that the Tychonoff product i∈I A (λ i ×λ i )×S i is a pseudocompact space. Then by Theorem 2.11 and Proposition 2.3.6 of [11] the map g : i∈I A (λ i × λ i ) × S i → i∈I B 0 λ i (S i ) defined by the formula g = i∈I g i is continuous, and hence the direct product B 0 λ i (S i ) : i ∈ I with the Tychonoff topology is a semiregular pseudocompact semitopological semigroup.…”

Section: Introductionmentioning

confidence: 99%

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Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

Гутік

Ravsky²

2017

J Math Sci

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In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt λ 0 i -extensions of semitopological monoids with zero. In particular we show that ifa family of Hausdorff pseudocompact topological Brandt λ 0 i -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product {S i : i ∈ I } is a pseudocompact space then the direct product B 0 λi (S i ), τ 0 B(Si) : i ∈ I endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup. Date: September 24, 2018. 2010 Mathematics Subject Classification. Primary 22A15, 54H10. Key words and phrases. Semigroup, Brandt λ 0 -extension, semitopological semigroup, topological Brandt λ 0 -extension, pseudocompact space, countably compact space, countably pracompact space, sequentially compact space, ω-bounded space, totally countably compact space, countable pracompact space, sequential pseudocompact space.

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On feebly compact inverse primitive (semi)topological semigroups

Cited by 3 publications

References 20 publications

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

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

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

Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

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