Sets of statistical cluster points and ℐ-cluster points.

Abstract

Let $\mathcal{I}$ be an admissible (i.e., proper and containing all finite subsets of $\mathbb{N}$) ideal of subsets of the set $\mathbb{N}^+$ of positive integers. The concept of $\mathcal{I}$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\mathcal{I}$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a boundedly compact metric space which ensures the connectedness of the set of all statistical cluster points of $x$. If $\Gamma_x(\mathcal{I})$ denotes the set of all $\mathcal{I}$-cluster points of the sequence $x$ and $M$ is a set of sequences in a metric space $X$ such that, for each $x\in M$, $\Gamma_x(\mathcal{I})\ne\emptyset$, then the assignment $x\mapsto\Gamma_x(\mathcal{I})$ gives a map $M\to\mathcal{F}$ where $\mathcal{F}$ is the set of all non-empty closed subsets of the space $X$ or $M\to\mathcal{C}$ where $\mathcal{C}$ is a suitable subset of $\mathcal{F}$. In the second part of the paper we study the continuity of this map with respect to the sup-metric on $M$ and some standard hypertopologies on $\mathcal{C}$ (the Vietoris topology, the Fell topology, the proximal topology and the topology given by the Hausdorff metric). We obtain some positive results in the case of locally compact and, particularly, boundedly compact metric spaces.