SOME COVERING SPACES AND TYPES OF COMPACTNESS

In this paper we shall study covering spaces such as fully normal spaces, absolutely countably compact, minimal Hausdor, @-space, realcompact, locally paracompact, w compact, maximal compact. Moreover, we give refinements of some theorems rasied in (1), also we shall give partial solutions of some open prob- lems raised in (2), and (3). In 1996 M. L. Puertas suggested the following question: If every proper subspace (A, A) of the space (X, ) has a property P, should the original space (X, ) have the property P? Nowadays, such kind of topological properties are known as properly hereditary properties. More precisely, a topological property is called a properly hereditary property if every proper subspace has the property, then the whole space has the property. Moreover, if every proper closed (open, F , G , etc.) subspace has the property, then the whole space has the property, we call such a property properly closed (open, F , G , etc.) hereditary. F. Arenas in (3) studied Puertas's problem and proved that topological prop- erties like separation axioms (T0,T1,T2,T3), separability, countability axioms, and metrizability are properly hereditary properties. At the end of his paper Arenas (3) suggested some open problems. Some of these problems were solved by Al- Bsoul in (1) and (2). Also, in (1) and (2), Al-Bsoul proved that many topological properties are properly hereditary properties. Moreover, Al-Bsoul suggested new open problems concerning this concept. In this paper some open problems raised in (2) and (3) will be solved. Moreover, we proved that the following topological properties are properly hereditary prop- erties: fully normality, absolutely countably compactness, locally paracompactness, minimal Hausdor, realcompactness , maximal compactness, @-space and !-space.