© Palestine Polytechnic University-PPU 2022.We introduced the concept of (a)-θ-compactness and (a)-θ-Mengerness in (a)topological spaces. We discussed the relationship of the above notions with the other known covering properties. It is shown that the product of two (a)-θ-Menger (resp. (a)-θ-compact) spaces is (a)-θ-Menger (resp. (a)-θ-compact) if one of them is (a)s-compact. If Xi is (a)-θ-Menger for each finite i, then (a)topological space X satisfies the selection principle Sfin(Θ-Ω(X ), Θ-Ω(X)). Further, it is shown that the (a)-θ-Menger covering property is preserved under (a)-θ-continuous and (a)-strongly-θ-continuous map.