  We define topological orthoalgebras (TOAs) and study their properties. While
every topological orthomodular lattice is a TOA, the lattice of projections of
a Hilbert space is an example of a lattice-ordered TOA that is not a toplogical
lattice. On the other hand, we show that every compact Boolean TOA is a
topological Boolean algebra. We also show that a compact TOA in which 0 is an
isolated point is atomic and of finite height. We identify and study a
particularly tractable class of TOAs, which we call {\em stably ordered}: those
in which the upper-set generated by an open set is open. This includes all
topological OMLs, and also the projection lattices of Hilbert spaces. Finally,
we obtain a topological version of the Foulis-Randall representation theory for
stably ordered TOAs
