This paper arose from the
following analogous questions: (1) Does a distributive topological lattice on a
continuum admit sufficiently many continuous lattice homomorphisms onto the unit
interval to separate points, and (2) does a topological semilattice on a continuum
admit sufficiently many continuous semilattice homomorphisms onto the unit
interval to separate points? Earlier investigations of topological lattices and
semilattices have provided partial positive solutions. However, examples of an
infinite-dimensional distributive lattice and a one-dimensional semilattice which
admit only trivial homomorphisms into the interval are presented in this
paper.
