Some properties of $\mathcal I$-convergence in cone metric spaces

Abstract

Let $\mathcal{I}$ be an ideal on $\mathbb N$, $\mathcal{I}$-sequential compactness, $\mathcal I$-sequentially countable compactness and $\mathcal{I}$-completeness in cone metric spaces are discussed. We also construct a bounded sequence in an infinitely discrete metric space which is not $\mathcal I$-convergent, which gives a negative answer to an open problem posed by P. Das \cite[Open problem 2.3]{Das}.