An Anderson–Choquet-type theorem and a characterization of weakly chainable continua

We introduce the concept of proper convergence of a sequence of subspaces of a metric space and then prove that a continuum X is weakly chainable if there is a sequence of arcs converging properly to it. Also, we prove that a continuum X is weakly chainable if and only if there is a sequence of arcs in the Hilbert cube converging properly to an embedded copy of X . The proof is based on an Anderson–Choquet-type theorem (valid also for set-valued functions).