Let $K$ be a valued field and $IK[[X]]$ the commutative algebra of integral functions over $K$. This paper is devoted to study some semigroups $S$ of $(IK[[X]],\circ)$, where $f\circ g$ is the composite function of $f,g\in IK[[X]]$. In the first section we define a. topology $\operatorname{Inv}_KS$ on $K$ and we extend to integral functions some notions used for polynomials (see.[5] and [6]) Here we study some connections between the subsemigroups $(S,\circ)$ of $(IK[[X]],\circ)$ and the topologies $\operatorname{Inv}_KS$ on $K$. In the second section we study when a particular subset of $K$ is an open set in the topology defined on $K$ by some semigroup of integral functions.