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

Hecker, Candyce; Millspaugh, Richard P.

Missouri J. Math. Sci. 21, No. 2, 78-82 (2009).

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.

Reviewer: Manuel Lopez Pellicer (Valencia)

MSC:

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

Keywords:

Closed set; metric space; separable subspace

Citations:

Zbl 1085.26002