To every directed graph E one can associate a graph inverse semigroup G(E), where elements roughly correspond to possible paths in E. These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C * -algebras, and Toeplitz C * -algebras. We investigate topologies that turn G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E) \ {0} must be discrete for any directed graph E. On the other hand, G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E, G(E) admits a T 1 semigroup topology in which G(E) \ {0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E) in larger topological semigroups.

We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

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