  A class of ordered relational topological spaces is described, which we call
orthomodular spaces. Our construction of these spaces involves adding a
topology to the class of orthomodular frames introduced by Hartonas, along the
lines of Bimb\'o's topologization of the class of orthoframes employed by
Goldblatt in his representation of ortholattices. We then prove that the
category of orthomodular lattices and homomorphisms is dually equivalent to the
category of orthomodular spaces and certain continuous frame morphisms, which
we call continuous weak p-morphisms. It is well-known that orthomodular
lattices provide an algebraic semantics for the quantum logic Q. Hence, as an
application of our duality, we develop a topological semantics for Q using
orthomodular spaces and prove soundness and completeness.
