On the set of all I-convergent sequences over different spaces

Abstract

In this article we elaborately study certain characteristics of the set of all $\mathcal{I}$-convergent sequences over various topological spaces. Earlier results of different authors were concerned regarding the closeness property of the sets: set of all bounded statistically convergent sequences, set of all bounded statistically convergent sequences of order $\alpha,$ set of all bounded $\mathcal{I}$-convergent sequences over the space $\ell^\infty$ ($\ell^\infty$- endowed with the sup-norm) only. On this context apart from this observation other properties (like connected and dense) of all three above mentioned sets have not yet been discussed over any other spaces. Our approach is to examine different behaviors of the set of all $\mathcal{I}$-convergent sequences over different spaces. Finally we are able to exhibit a condition over sequence spaces for which the set of all $\mathcal{I}$-convergent sequences form a closed set.