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On Topological Spaces Defined by \({\mathcal {I}}\)-Convergence

  • Xiangeng Zhou
  • Li Liu
  • Shou LinEmail author
Original Paper
  • 19 Downloads

Abstract

Ideal convergence in a topological space is induced by changing the definition of the convergence of sequences on the space by an ideal. Let \({\mathcal {I}}\subseteq 2^{\mathbb {N}}\) be an ideal. A sequence \((x_{n}:n\in {\mathbb {N}})\) in a topological space X is said to be \(\mathcal I\)-convergent to a point \(x\in X\) provided for any neighborhood U of x in X, we have the set \(\{n\)\(\in {\mathbb {N}}:x_{n}\notin U \}\in {\mathcal {I}}\). Recently, \({\mathcal {I}}\)-sequential spaces and \({\mathcal {I}}\)-Fréchet-Urysohn spaces are introduced and studied. In this paper, we discuss some topological spaces defined by \({\mathcal {I}}\)-convergence and their mappings on these spaces, expound their operation properties on these spaces, and study the role of maximal ideals of \({\mathbb {N}}\) in \(\mathcal I\)-convergence. We can apply \({\mathcal {I}}\)-convergence to unify and simplify the proofs of some old results in the literature and obtain some new results on the usual convergence and statistical convergence of topological spaces.

Keywords

Ideal convergence Statistical convergence \({\mathcal {I}}\)-sequential space \({\mathcal {I}}\)-Fréchet-Urysohn space \({\mathcal {I}}\)-continuous mapping Quotient mapping 

Mathematics Subject Classification

54A20 54B15 54C08 54D55 40A05 26A03 

Notes

Acknowledgements

The authors would like to thank the referee for the report of high quality. The report gives a series of essential comments to improve the original paper. For example, it considers the comparison of ideal convergence with respect to distinct ideals of \({\mathbb {N}}\) under the inclusion order, establishes some related results, discusses the role of maximal ideals of \({\mathbb {N}}\) in ideal convergence, introduces the class \(\Theta ({\mathcal {I}})\) and the space \(\Sigma ({\mathcal {J}})\), and brings out some questions for further study. In particular, the following results of this paper belong to the referee: Example 2.7, Definition 3.15, and Theorems 3.17 and 4.4.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsNingde Normal UniversityNingdePeople’s Republic of China

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