  We initiate a study of topological orthoalgebras (TOAs), concentrating on the
compact case. Examples of TOAs include topological orthomodular lattices, and
also the projection lattice of a Hilbert space. As the latter example
illustrates, a lattice-ordered TOA need not be a topological lattice. However,
we show that a compact Boolean TOA is a topological Boolean algebra. Using
this, we prove that any compact regular TOA is atomistic, and has a compact
center. We prove also that any compact TOA with isolated 0 is of finite height.
We then focus on stably ordered TOAs: those in which the upper-set generated by
an open set is open. These include both topological orthomodular lattices and
interval orthoalgebras -- in particular, projection lattices. We show that the
topology of a compact stably-ordered TOA with isolated 0 is determined by that
of of its space of atoms.
