Abstract: Some fundamental properties of topological inverse semigroups are discussed and it has been proved that the direct product/sum of a family of inverse semigroups is still an inverse semigroup and the semidirect product of two inverse semigroups is still an inverse semigroup under some certain conditions. In addition, it has been shown that in a given compact inverse semigroup, the closure of an inverse subsemigroup is a topological inverse subsemigroup, and the closure of a Clifford subsemigroup is a Clifford topological inverse subsemigroup. Thus the corresponding results related to topological semigroups or topological groups have been generalized.

Key words: topological inverse semigroup, semidirect product, topo logically closure