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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.