Continuous images of arcs: extensions of Cornette's theorem
A continuum is a compact connected Hausdorff space. A cyclic element in a locally connected continuum \(X\) is either (a) a connected subset \(M\) of \(X\) which is maximal with respect to the property that no point disconnects \(M\) or (b) a cutpoint of \(X\). A family \(G\) of sets in a continuum \(X\) is null if for each open cover \(W\) of \(X\) there are at most finitely many \(g \in G\) such that \(g\) is not contained in \(U\) for any \(U \in W\). It is well known that each cyclic element is a retract of \(X\); the family of cyclic elements of \(X\) is null and two distinct cyclic elements can intersect in at most one point. An arc is a continuum with a linear order such that its topology is given by the order topology. A continuum \(X\) is an IOC if it is the continuous image of an arc. Clearly, IOCs are locally connected continua and in the metric case, by the classical Hahn-Mazurkiewicz theorem, metric IOCs are the locally connected, metric continua. In 1960, \textit{S. Mardešić} [Proc. Am. Math. Soc. 11, 929--937 (1961; Zbl 0100.19003)] constructed a locally connected continuum which is not connected by arcs. \textit{J. L. Cornette} [Trans. Am. Math. Soc. 199, 253-267 (1974; Zbl 0291.54039)] proved that a locally connected continuum is an IOC if and only if each of its cyclic elements is an IOC. In the paper under review, the authors extend the result of Cornette by proving the following theorems. Theorem 2. Let \(X\) be a locally connected continuum and \(G\) be a null family which covers \(X\). Suppose that {\parindent=6mm \begin{itemize} \item[(i)] each \(g\) in \(G\) is an IOC (possibly degenerate), \item [(ii)] if \(h\), \(k\) are disjoint elements in \(G\), then there exists an element \(g\) of \(G\) and a closed subset \(A\) of \(g\) that separates \(h\) and \(k\) in \(X\), and \item [(iii)] for every \(g\), \(h\) in \(G\), \( g\smallsetminus h\) is contained in a component of \(X\smallsetminus h\). \end{itemize}} Then \(X\) is an IOC. Theorem 3. Let \(X\) be a locally connected continuum and \(G\) be a null family which covers \(X\). Suppose that {\parindent=6mm \begin{itemize} \item[(i)] each \(g\) in \(G\) is a continuous image of some compact ordered space with the order topology, \item [(ii)] if \(h\), \(k\) are disjoint elements in \(G\), then there exists an element \(g\) of \(G\) and a closed subset \(A\) of \(g\) that separates \(h\) and \(k\) in \(X\), and \item [(iii)] for every \(g\), \(h\) in \(G\), \(g\smallsetminus A\) is contained in a component of \(X\smallsetminus A\) for each closed subset \(A\) of \(h\). \end{itemize}} Then \(X\) is an IOC. The authors finish the paper with the following open problem. Question 1. Let \(X\) be a locally connected continuum and \(G\) a null cover of \(X\) satisfying (i) and (ii) in Theorem 2. Is \(X\) an IOC?
scientific article; zbMATH DE number 6506271
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