Introduction. Suppose that 5 is a topological semigroup which contains the bicyclic semigroup B as a subsemigroup. Let T denote the closure of B in 5. We investigate the structure of the semigroup T and the extent to which B determines this structure. In §1, two properties of F are established which hold for arbitrary 5; namely, that B is a discrete open subspace of F and T\B is an ideal of T if it is nonvoid. In §11, we introduce the notion of a topological inverse semigroup and establish several properties of such objects. Some questions are posed. In §111, it is shown that if 5 is a topological inverse semigroup, then T\B is a group with a dense cyclic subgroup. §IV contains a description of three examples of a topological semigroup which contains B as a dense proper subsemigroup. Finally, in §V, we assume that 5 is a locally compact topological inverse semigroup and show that either B is closed in 5 or F is isomorphic with the last of the examples described in §IV. A corollary about homomorphisms from B into a locally compact topological inverse semigroup is obtained which generalizes a result due to A. Weil [1, p. 96] concerning homomorphisms from the integers into a locally compact group. All spaces are topological Hausdorff in this paper. We state the definitions of Green's equivalence relations in a semigroup and the definition of an inverse semigroup. Green's relations £f, &, Jf, and 3i on a semigroup 5 are defined by : a@b if and only if a u aS=b\J bS, a£Cb if and only if a u Sa=b u Sb, JP=
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