Limit Sets and Closed Sets in Separable Metric Spaces

Abstract

Example 2 in Chapter 5 of [1] constructs, for an arbitrary closed subset of the real line, a sequence whose set of limit points is exactly the original closed set. We use a similar construction to show that an arbitrary nonempty closed set in a separable metric space is always the set of limit points of some sequence. We note further that if all nonempty closed subsets of a metric space can be realized as sets of limit points of sequences, then that metric space is separable.