Some topological properties of topological rough groups

Lin, Fucai; Sun, Qianqian; Lin, Yujin; Li, Jinjin

doi:10.1007/s00500-021-05631-6

Cited by 5 publications

(4 citation statements)

References 16 publications

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“…Definition 4.11. [16] A pre-topological group G is a group which is also a pre-topological space such that the multiplication mapping of G × G into G sending x × y into x • y, and the inverse mapping of H into G sending x into x −1 , are pre-continuous mappings. Definition 4.12.…”

Section: Proof For Each Covermentioning

confidence: 99%

“…Definition 4.12. [16] If a pre-topological G has a symmetric pre-base B e at the identity e such that for each U ∈ B e there exists V ∈ B e so that V 2 ⊂ U , then we say that G is a strongly pre-topological group. Theorem 4.13.…”

Section: Proof For Each Covermentioning

confidence: 99%

“…Further, J. Li first discuss the pre-topology (that is, the subbase for the topology) with the applications in the theory of rough sets, see [10,11,12], and then D. Liu in [13,14] discuss some properties of pre-topology. Recently, Lin, Cao and Li [15] have systematically investigated some properties of pre-topology, and Lin, Wu, Xie and Bao in [16] have introduced the concept of pre-topological group and studied some properties of it. Since the concept of uniformity plays an important role in the study of topological spaces, it is natural to pose the concept of pre-uniform structure as pre-topological tools in the applications of pre-topological spaces.…”

Section: Introductionmentioning

confidence: 99%

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Some properties of Pre-uniform spaces

Lin¹,

Xie²,

Wu³

et al. 2022

Preprint

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In this paper, we introduce the notions of pre-uniform spaces and preproximities and investigate some basic properties about them. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition of a pre-base, a family of strongly pre-uniform covers, or a family of strongly pre-uniform pseudometrics. As an application, we show that each strongly pre-topological group is completely regular. Finally, we pose the concept of the pre-proximity on a set and discuss some properties of the pre-proximity.

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Section: Proof For Each Covermentioning

confidence: 99%

Section: Proof For Each Covermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some properties of Pre-uniform spaces

Lin¹,

Xie²,

Wu³

et al. 2022

Preprint

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“…x → x −1 , is also continuous. The the properties of topological groups have been widely used in the study of topology, analysis and category, see [1,2,4,27,28,29,30,31,32,38,39]. For more details about topological groups, the reader see [3].…”

Section: Introductionmentioning

confidence: 99%

Some properties of Pre-topological groups

Lin¹,

Wu²,

Xie³

et al. 2022

Preprint

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In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each T 0 pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups.

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On separation axioms of topological rough groups

Liu²,

Mou³

et al. 2022

Soft Comput

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Some topological properties of topological rough groups

Cited by 5 publications

References 16 publications

Some properties of Pre-uniform spaces

Some properties of Pre-uniform spaces

Some properties of Pre-topological groups

On separation axioms of topological rough groups

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