Each regular paratopological group is completely regular

Банах, Тарас; Ravsky, Alex

doi:10.1090/proc/13318

Cited by 44 publications

(20 citation statements)

References 13 publications

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“…Since every C-embedded subspace of a regular Lindelöf is pseudo-ω 1 -compact [ Proof. Since every regular paratopological group is a Tychonoff space [3,Corollary 5], and every Tychonoff R-factorizable paratopological group is totally ω-narrow, i.e. the topological group G * associated to G is ω-narrow, the c(G * ) ≤ 2 ω [1, Theorem 5.4.10] and furthermore, G as a continuous image of satisfies that c(G) ≤ c(G * ) ≤ 2 ω .…”

Section: Completions Of Simply Sm-factorizable Topological Groupsmentioning

confidence: 99%

Simply sm-factorizable (para)topological groups and their completions

Xie

Tkachenko

2020

Monatsh Math

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In this paper, firstly, we give some characterizations of simply sm-factorizable (para)topological groups by continuous real-valued functions. It mainly shows that G is a simply sm-factorizable (para)topological group if and only if, for each continuous function f : G → R, there are a sub-sm-(para)topological group H, a continuous homomorphism h of G onto H and a continuous function g :Secondly, applying those results we study the completions of simply sm-factorizable topological groups. We prove that the equalities µG = ̺ω(G) = υG hold for each Hausdorff simply sm-factorizable topological group G. This result gives a positive answer to a question posed by Arhangel'shiǐ and Tkachenko [2, Problem 7.6]. Finally, we discuss the realcompactification of simply sm-factorizable paratopological groups. We mainly, among other results, the realcompactification υG of a regular simply smfactorizable paratopological group admits a natural structure of paratopological group containing G as a dense subgroup and υG is also simply sm-factorizable. Some results in [12] are improved.2010 Mathematics Subject Classification. 22A05, 22A30,54H11, 54A25, 54C30.

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Section: Completions Of Simply Sm-factorizable Topological Groupsmentioning

confidence: 99%

Simply sm-factorizable (para)topological groups and their completions

Xie

Tkachenko

2020

Monatsh Math

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“…Basic separation axioms and relations between them are considered in [Eng,Section 1.5]. For more specific cases and topics, also related to semitopological and paratopological groups, see [Rav2], [BanRav2], [Tka,Section 2], [Tka2], [XLT].…”

Section: Preliminariesmentioning

confidence: 99%

A note on compact-like semitopological groups

Ravsky

2019

Carpathian Math. Publ.

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The note contains a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is presented a semiregular semitopological group G which is not T 3 . We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular T 1 compact and T 2 sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group (G, τ ) is a topological group provided there exists a Hausdorff topology σ ⊃ τ on G such that (G, σ) is a precompact topological group and (G, τ ) is weakly semiregular or (G, σ) is a feebly compact paratopological group and (G, τ ) is T 3 .

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“…Then G = i∈I G i is a Baire precompact paratopological group. Hence the topology of (G i ) * forms a π-base for G i (see [4,Theorem 5]). Apply Lemma 3.1 to conclude that G * ∼ = i∈I (G i ) * , where each (G i ) * is a Baire precompact topological group by Lemma 3.2.…”

Section: Products Of Paratopological Groupsmentioning

confidence: 99%

“…Apply Lemma 3.1 to conclude that G * ∼ = i∈I (G i ) * , where each (G i ) * is a Baire precompact topological group by Lemma 3.2. Hence, according to [4,Theorem 3.4], G * is a Baire precompact topological group. Since G is precompact and the topology of G * is a π-base for G, Lemma 3.2 implies that G is Baire.…”

Section: Products Of Paratopological Groupsmentioning

confidence: 99%

Baire property in product spaces

2015

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We show that if a product space Π has countable cellularity, then a dense subspace X of Π is Baire provided that all projections of X to countable subproducts of Π are Baire. It follows that if Xi is a dense Baire subspace of a product of spaces having countable π-weight, for each i ∈ I, then the product space i∈I Xi is Baire. It is also shown that the product of precompact Baire paratopological groups is again a precompact Baire paratopological group. Finally, we focus attention on the so-called strongly Baire spaces and prove that some Baire spaces are in fact strongly Baire.2010 MSC: 54H11; 54E52.

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Each regular paratopological group is completely regular

Cited by 44 publications

References 13 publications

Simply sm-factorizable (para)topological groups and their completions

Simply sm-factorizable (para)topological groups and their completions

A note on compact-like semitopological groups

Baire property in product spaces

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