Function sequences, uniform convergence, and statistical aspects in bipolar metric convergence

Reha Yapalı; Utku Gürdal; Reha Yapalı; Utku Gürdal

doi:10.3934/math.20261230

AIMS Mathematics

2026, Volume 11, Issue 9: 31113-31133. doi: 10.3934/math.20261230

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Research article

Function sequences, uniform convergence, and statistical aspects in bipolar metric convergence

Reha Yapalı ^{1
,
,},
Utku Gürdal ²

1.
Department of Mathematics, Muş Alparslan University, Muş, Türkiye
2.
Department of Mathematics, Burdur Mehmet Âkif Ersoy University, Burdur, Türkiye

Received: 14 July 2026 Revised: 05 September 2026 Accepted: 15 September 2026 Published: 22 September 2026
MSC : 40A35, 54C30, 54E35

In this paper, we first develop statistical convergence in bipolar metric spaces. This convergence is characterized both by the corresponding real-valued distance sequence and by the existence of a convergent subsequence indexed by a density-one set. We also show that convergence implies statistical convergence, establish uniqueness when the statistical limit is central, and obtain statistical sequential characterizations of continuity for mappings and comappings. For sequences of mappings, we introduce pointwise, uniform, pointwise statistical, and uniform statistical convergence, and prove that the uniform statistical limit of continuous mappings is a continuous comapping. The mapping–comapping duality thus provides a natural function-space interpretation of these limits and yields a bipolar counterpart of the classical uniform-limit theorem for continuous functions.
- bipolar metric space,
- uniform convergence,
- statistical convergence,
- bisequence,
- comapping
Citation: Reha Yapalı, Utku Gürdal. Function sequences, uniform convergence, and statistical aspects in bipolar metric convergence[J]. AIMS Mathematics, 2026, 11(9): 31113-31133. doi: 10.3934/math.20261230

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Abstract

In this paper, we first develop statistical convergence in bipolar metric spaces. This convergence is characterized both by the corresponding real-valued distance sequence and by the existence of a convergent subsequence indexed by a density-one set. We also show that convergence implies statistical convergence, establish uniqueness when the statistical limit is central, and obtain statistical sequential characterizations of continuity for mappings and comappings. For sequences of mappings, we introduce pointwise, uniform, pointwise statistical, and uniform statistical convergence, and prove that the uniform statistical limit of continuous mappings is a continuous comapping. The mapping–comapping duality thus provides a natural function-space interpretation of these limits and yields a bipolar counterpart of the classical uniform-limit theorem for continuous functions.

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