Countable Paracompactness and Weak Normality Properties By

After proving this theorem, we obtain similar results for the topological spaces studied in [7] and [11]. Also, cogent examples are given and the relation this note bears to the work of others is discussed. We shall follow the terminology of [5] except we shall assume separation properties for a space only when these assumptions are explicitly stated. For an infinite cardinal m, a set A in a topological space will be called a Gm-set (respectively, a regular Gm-set) provided it is the intersection of at most m open sets (respectively, at most m closed sets whose interiors contain A). If m = S0, we shall use the familiar terms GVset and regular C-set. It is clear that the zero-set of any continuous real valued function is a regular (/¿-set and that the intersection of no more than in such zero-sets is a regular Cmset. In the remaining part of this paper, we shall use these facts without explicitly mentioning them. Definition. For an infinite cardinal nt, a topological space is m-normal if each pair of disjoint closed sets, one of which is a regular Cm-set, have disjoint neighborhoods. For m = S0, we shall use the more suggestive term 8-normal. Note that a normal space is m-normal and that a regular space is normal if and only if it is m-normal for every infinite cardinal m. On the other hand, a compact Fi-space that is not Hausdorff is m-normal for every infinite cardinal but yet it fails to be normal. Recall that a space is m-paracompact if each open cover having cardinal less than or equal to m has a locally finite open refinement. Characterizations of mparacompact spaces may be found in [14] and [8].