Fuzzifying completeness and compactness in fuzzifying bornological linear spaces

Zhenyu Jin; Conghua Yan; Zhenyu Jin; Conghua Yan

doi:10.3934/math.2022916

AIMS Mathematics

2022, Volume 7, Issue 9: 16706-16718. doi: 10.3934/math.2022916

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Theory article

Fuzzifying completeness and compactness in fuzzifying bornological linear spaces

Zhenyu Jin ¹,
Conghua Yan ^{2
,
,}

1.
School of Mathematical Sciences, Suzhou University of Science and Technology, Jiangsu 215009, China
2.
School of Mathematical Sciences, Nanjing Normal University, Jiangsu 210023, China

Received: 03 May 2022 Revised: 28 June 2022 Accepted: 06 July 2022 Published: 12 July 2022
MSC : 46S40, 54A40

The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete.
Citation: Zhenyu Jin, Conghua Yan. Fuzzifying completeness and compactness in fuzzifying bornological linear spaces[J]. AIMS Mathematics, 2022, 7(9): 16706-16718. doi: 10.3934/math.2022916

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Abstract

The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete.

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