On the Rank of the Semigroup TE(X)

${cal T}_X $ denotes the full transformation semigroup on a set $ X $. For anontrivial equivalence $E$ on $X$, let[T_E (X) ={ fin {cal T}_X : forall , (a,b)in E,, (af,bf)in E } .]Then $T_E (X) $ is exactly the semigroup of continuous selfmaps ofthe topological space $X$ for which the collection of all$E$-classes is a basis. In this paper, we first discuss the rankof the homeomorphism group $G$, and then consider the rank of $T_E(X)$ for a special case that the set $X$ is finite and that eachclass of the equivalence $E$ has the same cardinality. Finally,the rank of the closed selfmap semigroup $Gamma(X)$ of the space$X$ is observed. We conclude that the rank of $G$ is no more than4, the rank of $T_E (X)$ is no more than 6 and the rank of$Gamma(X)$ is no more than 5.