Topological Lumpiness and Topological Extreme Amenability

In this paper we give some characterizations of topological extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative topological left invariant mean then T is topological left lumpy if and only if there is a multiplicative topological left invariant mean M on M(S)* such that M(?T)=1, where ?T is the characteristic functional of T. Consequently if T is a topological left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely topological left amenable if and only if S is.