A topological semigroup is a
Hausdorff space S together with a continuous associative multiplication
m : S ×S → S. The lifting of the group structure of a topological group to its simply
connected covering space is a technique used in the theory of Lie groups. In this
paper we investigate the lifting of the multiplication of a topological semigroup S to
its simply connected covering space (S,φ). A general theory is developed and
applications to examples are discussed.