Convergence properties of neighboring sequences

Abstract:

First countable spaces and semimetrizable spaces are examples of topological spaces which can be characterized in terms of convergence properties of sequences “neighboring” a point. In this paper we consider conditions sufficient for metrizability of spaces defined in terms of convergence properties of “neighboring” sequences, in particular, those in which the set of cluster points of one sequence is a subset of that of any “neighboring” sequence. The special case in which the sets of cluster points are equal is shown to be a characterization of metrizability in ${T_0}$-spaces.