Local selections and local dendrites
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 \(C(X,A,\epsilon)\neq \emptyset\).
scientific article; zbMATH DE number 3927886
scientific article