The set of compact connected metric spaces (continua) can be divided into classes according to the complexity of their descriptions as inverse limits of polyhedra. The simplest such class is the collection of chainable continua, i.e. those which are inverse limits of arcs. In 1964, A. Lelek introduced a notion which is related to chainability, called span zero. A continuum X has span zero if any two continuous maps from any other continuum to X with identical ranges have a coincidence point. Lelek observed that every chainable continuum has span zero; he later asked whether span zero is in fact a characterization of chainability. In this thesis, we construct a non-chainable continuum in the plane which has span zero, thus providing a counterexample for what is now known as Lelek's Problem in continuum theory. Moreover, we show that the plane contains an uncountable family of pairwise disjoint copies of this continuum. We discuss connections with the classical problem of determining up to homeomorphism all the homogeneous continua in the plane.

The set of compact connected metric spaces (continua) can be divided into classes according to the complexity of their descriptions as inverse limits of polyhedra. The simplest such class is the collection of chainable continua, i.e. those which are inverse limits of arcs. In 1964, A. Lelek introduced a notion which is related to chainability, called span zero. A continuum X has span zero if any two continuous maps from any other continuum to X with identical ranges have a coincidence point. Lelek observed that every chainable continuum has span zero; he later asked whether span zero is in fact a characterization of chainability. In this thesis, we construct a non-chainable continuum in the plane which has span zero, thus providing a counterexample for what is now known as Lelek's Problem in continuum theory. Moreover, we show that the plane contains an uncountable family of pairwise disjoint copies of this continuum. We discuss connections with the classical problem of determining up to homeomorphism all the homogeneous continua in the plane.