  A topological group is locally pseudocompact if it contains a non-empty open
set with pseudocompact closure. In this note, we prove that if G is a group
with the property that every closed subgroup of G is locally pseudocompact,
then G_0 is dense in the component of the completion of G, and G/G_0 is
zero-dimensional. We also provide examples of hereditarily disconnected
pseudocompact groups with strong minimality properties of arbitrarily large
dimension, and thus show that G/G_0 may fail to be zero-dimensional even for
totally minimal pseudocompact groups.
