Abstract. A T 1 topological space X is called an almost GP-space if every dense G δ -set of X has nonempty interior. The behaviour of almost GP-spaces under taking subspaces and superspaces, images and preimages and products is studied. If each dense G δ -set of an almost GP-space X has dense interior in X, then X is called a GID-space. In this paper, some interesting properties of GID-spaces are investigated. We will generalize some theorems that hold in almost P-spaces.