Principal topologies and transformation semigroups

For a given set X, the set F ( X ) of all maps from X to X forms a semigroup under composition. A subsemigroup S of F ( X ) is said to be saturated if for each x ∈ X there exists a set O x ⊆ X with x ∈ O x such that S = { f ∈ F ( X ) | f ( x ) ∈ O x ∀ x ∈ X } . It is shown that there exists a one-to-one correspondence between principal topologies on X and saturated subsemigroups of F ( X ) . Some properties of principal topologies on X and the corresponding properties of their associated saturated subsemigroups of F ( X ) are discussed.