Simply connected compact subsets of the plane
type: Item
instance of: scholarly article
title: Simply connected compact subsets of the plane
zbMATH Open document ID: 0938.54033
author: T. I. Ramsamujh
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-12-05T00:00:00Z
review text: For a compact subset \(A\) of the plane, it is said that \(A\) is a set with no holes provided that whenever \(\lambda\) is a Jordan curve in \(A\) we have that \(\operatorname {Ins}(\lambda) \subseteq A\). A simply connected set is a path-connected set with no holes. Denote by NH and SC the collections of all sets with no holes and all simply connected sets, respectively. In this paper the author presents a rank function for the set NH which measures the complexity of sets by looking at trees of sequences of Jordan arcs which in some sense approximate holes. He presents properties of the rank functions, characterizes the sets of rank 1, and shows that the rank function is unbounded in \(\omega_1\) on the set SC. The last result is then used to show that SC and NH can not be analytic sets in the complete separable metric space \(K(\mathbb{R}^2)\) of all compact subsets of the plane with the Hausdorff metric.
reviewed by: Sheldon W. Davis
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 1348146
zbMATH Keywords: rank function
zbMATH Keywords: complexity of sets
zbMATH Keywords: Jordan arcs
zbMATH Keywords: analytic sets
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6476
OpenAlex ID: W2074369428
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DOI: 10.1006/JMAA.1999.6476
description: scientific article; zbMATH DE number 1348146