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.

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关于$$ {\ mathcal {I}} $$ I定义的拓扑空间-收敛

拓扑空间中的理想收敛是通过理想改变空间序列的收敛性的定义而引起的。让\（{\ mathcal {I}} \ subseteq 2 ^ {\ mathbb {N}} \）是理想的。拓扑空间X中的序列\（（x_ {n}：n \ in {\ mathbb {N}}）\）被称为\（\ mathcal I \）-收敛到点\（x \ in X \）规定的任何邻域ü的X在X，我们有集\（\ {N \）\（\在{\ mathbb {N}}：X_ {N} \ notin U \} \在{\ mathcal { I}} \）。最近，\（{\ mathcal {I}} \）-连续空格和\（{\ mathcal {I}} \}-Fréchet-Urysohn空间得到了介绍和研究。在本文中，我们讨论由\（{\ mathcal {I}} \）-收敛定义的一些拓扑空间及其在这些空间上的映射，阐述它们在这些空间上的操作性质，并研究\（{ \ mathbb {N}} \）在\（\ mathcal I \） -convergence。我们可以应用\（{\ mathcal {I}} \）- convergence来统一和简化文献中一些旧结果的证明，并获得有关拓扑空间通常收敛和统计收敛的一些新结果。