Limit sets and closed sets in separable metric spaces
type: Item
instance of: scholarly article
title: Limit sets and closed sets in separable metric spaces
zbMATH Open document ID: 1209.54012
author: Candyce Hecker
author: Richard P. Millspaugh
published in: Missouri Journal of Mathematical Sciences
publication date: 2009-07-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 5575884
zbMATH Keywords: Closed set
zbMATH Keywords: metric space
zbMATH Keywords: separable subspace
reviewed by: M. Pellicer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5575884