On $$\mathcal {I}$$-neighborhood Spaces and $$\mathcal {I}$$-quotient Spaces

Abstract

An ideal on $$\mathbb N$$ is a family of subsets of $$\mathbb N$$ closed under the operations of taking finite unions and subsets of its elements. The $$\mathcal {I}$$ -open sets of topological spaces, which are determined by an ideal $$\mathcal {I}$$ on $$\mathbb N$$ and the topology of the spaces, are a basic concept of ideal topological spaces. However, it encounters some difficulties in the study of certain structures and mappings of topological spaces. In this paper, we discuss some properties of ideal topological spaces based on $$\mathcal {I}_{sn}$$ -open sets, study the problem generating new topological spaces from ideals, characterize the mappings preserving $$\mathcal {I}$$ -convergence and structure special $$\mathcal {I}$$ -quotient spaces. The following main results are obtained. These show the unique role of $$\mathcal {I}$$ -neighborhood spaces in the study of ideal topological spaces and present a version using the notion of ideals.