In this paper we give a topological representation of distributive meet-semilattice with last element 1. We study the notion of irreducible, weakly-irreducible filter and order-ideal in a meet-semilattice. We show a characterization of distributive semilatices by means of weakly-irreducible filters. These results are applied to give a topological representation by means of ordered topological spaces. Finally, we show a duality between meet-hemimorphism of distributive meet-semilattices and certain relations. In particular, we obtain a duality for homomorphism of distributive meet-semilattices by means of binary relations.