Study of convergence properties of sequences and series in certain metric spaces

Articles in Press

Document Type : Research Paper

Authors
Mojgan Javahernia 1
Rana Hajipouri Basmenj 2
1 Department of Mathematics, Shab.C., Islamic Azad University, Shabestar, Iran
2 Department of Mathematics, He.C., Islamic Azad University, Heris, Iran
10.22075/ijnaa.2026.40083.5603
Abstract
This paper investigates the convergence of sequences in metric-like spaces, a class of structures that generalize both metric and partial metric spaces. It begins with a review of the fundamental notions of quasimetric spaces, statistical convergence, ideal convergence, and $\mathcal{I}^*$-convergence. The concepts of $\mathcal{I}$- and $\mathcal{I}^*$-convergence for sequences in pseudo-metric spaces are then introduced, and their relationships are examined through rigorous analysis and carefully constructed counterexamples. The results establish that classical convergence always implies $\mathcal{I}$-convergence, although the converse does not necessarily hold. By employing specific ideals, such as sets of zero or finite natural density, the paper constructs various illustrative families of sequences. It is shown that if a sequence is $\mathcal{I}$-convergent, one can extract a subsequence that converges in the classical sense. Furthermore, $\mathcal{I}^*$-convergence is strictly stronger than $\mathcal{I}$-convergence, and every $\mathcal{I}^*$-convergent sequence is automatically $\mathcal{I}$-convergent. In $T_0$ spaces, or when the limit point is separated, the two modes of convergence exhibit a closer relationship. The AP condition on the ideal plays a central role in ensuring the equivalence between $\mathcal{I}$- and $\mathcal{I}^*$-convergence. Overall, the paper demonstrates that extending convergence notions in pseudo-metric spaces via ideals offers a robust analytical framework for understanding convergence in generalized metric structures.
Keywords
Convergence of sequences Quasimetric space
Ideal convergence
Condition (AP)
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International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 14 October 2026

  • Receive Date 16 December 2025
  • Revise Date 14 April 2026
  • Accept Date 20 April 2026