map Lindelof and some properties of space

Abstract

Let be a topological space , we say that is iff every Lindelof subspace of is closed in. In this paper we will present some properties of the Lindelof space, and then introduce some concepts that help us to prove some of the properties of. And we introduce a new concept of K- map Lindelof , then we find the necessary conditions that make each K-map Lindelof is k-map. we find the conditions on the function that make the direct image of is , and finally we have found the conditions on the function that make the inverse image of is ,with the knowledge that the two issues are not investigated in general.