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

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 $${\mathcal {I}}$$-Frechet-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.