An Anderson-Choquet-type theorem and a characterization of weakly chainable continua (Q2362989)

A finite sequence \({U_{1},\ldots,U_{n}}\) of subsets of a metric continuum \(X\) is a weak chain if for each \(i < n\), the set \(U_{i}\) intersects \(U_{i+1}\). The definition of weak chainability of \(X\) is similar to that of chainability but using weak chains instead of chains. It is known that the following are equivalent: (a) \(X\) is weakly chainable, (b) \(X\) is a continuous image of a chainable continuum, (c) \(X\) is a continuous image of the pseudo-arc, and (d) \(X\) admits a uniformizable approximation by arcs. In the paper under review, the authors prove an Anderson-Choquet-type theorem (valid also for set-valued functions) which gives as a result an onto mapping instead of a homeomorphism. Using this result they give a new characterization of weakly chainable continua, namely, a subcontinuum \(X\) of the Hilbert cube \(Q\) is weakly chainable if and only if there is a sequence of arcs in \(Q\) that converges properly to \(X\). The notion of proper convergence is introduced in this paper and the authors discuss the differences between their characterization and the property (d) above.