Limit sets and closed sets in separable metric spaces. (English) Zbl 1209.54012

In Chapter 5 of the book of [B. R. Gelbaum and 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.


54D65 Separability of topological spaces
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)


Zbl 1085.26002