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and definitions. Let S be a set of points. By an nsubset of S we shall mean a subset of S consisting of precisely n points. Let S be a subset of S and G a group of 1-1 transformations of S upon itself. Then G is weakly n-transitive over S if for any two n-subsets Xi, X2 of S there exists a gGG such that g(Xl) = X2. (It is not required that g(S) CS.) In particular if S= S, S is a topological space, and G is the group of homeomorphisms of S upon itself, then the space S is said to be weakly n-homogeneous.1 The purpose of this paper is to prove that an infinite weakly nhomogeneous topological space is weakly (n 1)-homogeneous (n> 1).

Weak 𝑛-homogeneity implies weak (𝑛-1)-homogeneity