  We introduce the notion of orthogonality in a vector space with a topology on
it. To serve our purpose, we define orthogonality space for a given vector
space X, using the topology on it. We show that for a suitable choice of
orthogonality space, Birkhoff-James orthogonality in a Banach space is a
particular case of the orthogonality introduced by us. We characterize the
right additivity of orthogonality in our setting and obtain a necessary and
sufficient condition for a Banach space to be smooth as a corollary to our
characterization. Finally, using our notion of orthogonality, we obtain a
topological generalization of the Bhatia-Semrl Theorem.
