Countable paracompactness and weak normality properties

In [4], Dowker proved that a normal space X is countably paracompact if and only if its product with the closed unit interval is normal.In this paper, we prove an analogue of Dowker's theorem.Specifically, we define the term 8-normal and then prove the following: Theorem 1.A topological space is countably paracompact if and only if its product with the closed unit interval is 8-normal.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].Theorem 2. Each m-paracompact space is m-normal.