The principal results are the
following. If M is a metric space homeomorphic to a subset of a real linear space
which is star-shaped with respect to an element p, or if M is homeomorphic to an
arcwise connected subspace of a dendroid which is smooth at a point p, then each
closed subset of M which contains p is the fixed point set of a continuous mapping of
M. If M is a continuum having Property W (this is a class of Peano continua
containing the local dendrites and the continua containing no continuum
of condensation) then each nonempty closed subset of M is a fixed point
set. It is shown that a subset K of a dendrite is the fixed point set of a
continuous surjection if and only if the complement of K is not homeomorphic to
[0,∞).
