Limit sets and closed sets in separable metric spaces (Q1026903)

In Chapter 5 of the book of [\textit{B. R. Gelbaum} and \textit{J. M. H. Olmstead}, Counterexamples in Analysis. Mineola, NY: Dover Publications. (2003; Zbl 1085.26002)], the authors construct for an arbitrary closed subset of the real line a sequence whose set of limit points is exactly the original set. In this paper a nice similar construction is done to prove that an arbitrary nonempty closed set in a separable metric space is always the set of limit points of some sequence. The authors note that if all nonempty closed subsets of a metric space can be realized as sets of limit points of sequences, then the metric space is separable.