## Departmental PhD Thesis Exam – Logan Hoehn

Tuesday, June 29, 2010, 2:00 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Logan Hoehn

Thesis Title: Non-chainable continua and Lelek's problem

Thesis Abstract:
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 thesis can be found at
http://www.math.utoronto.ca/lhoehn/Thesis.pdf