Local connectivity functions on arcwise connected spaces and certain continua
This is a continuation of the author's paper [Topol. Proc. 28, 229--239 (2004; Zbl 1081.54030)]. In the paper under review, the author adds a statement (part (1) in the following theorem) to the main theorem in the cited paper: Theorem 7. For a continuum \(X\), the following three statements are equivalent: (1) every connected real-valued function on \(X\) is a local connectivity function; (2) every connected real-valued function on \(X\) is a connectivity function; (3) every connected real-valued function on \(X\) is a Darboux function; (4) \(X\) is a dendrite such that each arc in \(X\) contains only finitely many branch points of \(X\). The author introduces a nice concept called \(\varepsilon\)-splitting set, and uses it to prove the following characterizations: Theorem 13. Let \(X\) be an arc-like continuum. Then the following statements are equivalent: (1) every real-valued local connectivity function on \(X\) is a connected function; (2) every real-valued local connectivity function on \(X\) is a Darboux function; (3) every real-valued local connectivity function on \(X\) is a connectivity function; (4) \(X\) is an arc. Theorem 17. Let \(X\) be a circle-like continuum. Then the following statements are equivalent; (1) every real-valued local connectivity function on \(X\) is a Darboux function; (2) every real-valued local connectivity function on \(X\) is a connectivity function; (3) \(X\) is a simple closed curve. A key result of this paper, which is used in most of the results is: Lemma 1. Let \(X\) be a Peano continuum. Then every local connectivity function from \(X\) to a topological space \(Y\) is a connected function. With these results the author gives partial answers to questions of J. Stallings, posed in 1959.
scientific article; zbMATH DE number 5013540