Embedding one-dimensional continua into \(T\times I\) (Q860486)

A motivation of this paper is a question of W. Lewis asking whether given a locally planar one-dimensional continuum \(M\), there exists an embedding \(\varphi\) of \(M\) into the product of a simple triod and and arc (abbrivated by \(T\times I)\). In this paper the author considers the more general problem of embedding one-dimensional continua into \(T\times I\). A collection of open sets \({\mathcal C}=\{D_{\alpha}\}_{\alpha}\in A\) is said to be a graph-chain provided that \({\mathcal A}\) is a finite partially ordered set such that (a) \(\alpha\) is adjacent to \(\beta\) if and only if \(D_{\alpha}\cap D_{\beta}\neq \emptyset\), and (b) if \(\alpha\) and \(\beta\) are two elements of \({\mathcal A}\) that are each adjacent to more than two elements of \({\mathcal A}\), then \(D_{\alpha}\cap D_{\beta}= \emptyset\). A continuum \(M\) is said to be graph-chainable provided that for each \(\varepsilon >0\) there exists a graph-chain of mesh less than \(\varepsilon\) that covers \(M\). A graph is said to be piecewise linear if all of its chain components are linear. Given a continuum \(M\), a subset \(N\) of \(M\) is called expandable if there exists \(S\subset M\) such that \(S\) is planar, and the closure of \(N\) is contained in \(S\). The main result is: Let \(M\) be a graph-chainable continuum such that there exists a finite family \({\mathcal F}\) of pairwise disjoint planar neighborhood of \(M\) such that \(M\setminus \bigcup {\mathcal F}\) is piecewise linearly chainable. Then there exists an embedding of \(M\) into \(T\times I\) provided that one of the following properties is satisfied: (a) for each \(S\in {\mathcal F}\), the diameter of \(S\) is less than the supremum of real numbers \(\delta\) such that if the diameter of \(N\subset M\) is less than \(\delta\), then \(N\) is planar, or (b) each \(S\in {\mathcal F}\) is expandable.