Local selections and local dendrites
type: Item
instance of: scholarly article
title: Local selections and local dendrites
zbMATH Open document ID: 0579.54025
DOI: 10.1016/0166-8641(85)90034-3
author: L. E. jun. Ward
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Given a continuum X and \(\epsilon >0\) the symbol \(C(X)\) (\(C(X,\epsilon)\)) denotes the space of all subcontinua of X (of diameter less than \(\epsilon)\). A selection on some subspace of C(X) is a continuous choice function; the selection \(\sigma\) is rigid if \(\sigma(A)\in B\subset A\) implies \(\sigma (A)=\sigma (B)\). The first part of the paper is devoted to the existence of selections and rigid selections on whole C(X) and contains the known result that a continuum admits a rigid selection iff it is a smooth dendroid [the author, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys. 19, 1041-1044 (1971; Zbl 0221.54011)]. The second one is devoted to the existence of selections and rigid selections on \(C(X,\epsilon)\). It is proved that a Peano continuum X is a local dendrite (and contains at most one simple closed curve) iff \(C(X,\epsilon)\) admits a selection (a rigid one) for some \(\epsilon >0.\) In the third part the author discusses the existence of local selections at some members of C(X). It is said that C(X) admits a local selection at A if there is a selection on \(C(X,A,\epsilon) = \{B\in C(X): H(A,B)<\epsilon\}\) for some \(\epsilon >0\) (H denotes the Hausdorff distance). It is proved that for a Peano continuum X and for \(A\in C(X)\) the following conditions are equivalent: (i) C(X) admits a local selection at A; (ii) C(X) admits a rigid local selection at A; (iii) C(X) admits a constant local selection at A; (iv) there is \(\epsilon >0\) such that \(C(X,A,\epsilon)\neq \emptyset\).
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 3927886
zbMATH Keywords: \(\theta \) -curve
zbMATH Keywords: web
zbMATH Keywords: cyclic continuum
zbMATH Keywords: rigid selections
zbMATH Keywords: smooth dendroid
zbMATH Keywords: Peano continuum
zbMATH Keywords: local dendrite
zbMATH Keywords: local selections
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90034-3
OpenAlex ID: W2044110309
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description: scientific article; zbMATH DE number 3927886