1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T
1 space X for which
forms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].
The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.