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<p>In recent years, important results have emerged from the study of convergence theory concepts in generalized metric structures. In this work, within the framework of <i>A</i>-metric spaces (<i>X</i>,&#xa0;<i>A</i>), which are a particular example of generalized metric spaces, ideal <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(I_A\)</EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(I_A^*\)</EquationSource> </InlineEquation>-Cauchy sequences, as well as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(I_A\)</EquationSource> </InlineEquation>- and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(I_A^*\)</EquationSource> </InlineEquation>-divergent sequences, are defined. Under suitable conditions, it is shown that every <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(I_A^*\)</EquationSource> </InlineEquation>-Cauchy sequence is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(I_A\)</EquationSource> </InlineEquation>-Cauchy, and that the (AP) condition is necessary and sufficient for the equivalence between Cauchy and divergent sequences. This study generalizes these concepts from previous metric space results to <i>A</i>-metric spaces, providing the first systematic definition and analysis of such sequences in this setting.</p>

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A Note on ideal Cauchy Sequences and Ideal Divergence in A-Metric Spaces

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Ramazan Sunar,
  • Mukaddes Arslan

摘要

In recent years, important results have emerged from the study of convergence theory concepts in generalized metric structures. In this work, within the framework of A-metric spaces (X, A), which are a particular example of generalized metric spaces, ideal \(I_A\) - and \(I_A^*\) -Cauchy sequences, as well as \(I_A\) - and \(I_A^*\) -divergent sequences, are defined. Under suitable conditions, it is shown that every \(I_A^*\) -Cauchy sequence is \(I_A\) -Cauchy, and that the (AP) condition is necessary and sufficient for the equivalence between Cauchy and divergent sequences. This study generalizes these concepts from previous metric space results to A-metric spaces, providing the first systematic definition and analysis of such sequences in this setting.