Characterizations of Relatively Countably Paracompact Spaces

We shall introduce the relatively countably n-paracompactness in a total space (n = 1,2,3) and discuss the relations among another relative properties, and also give their char-acterizations. AMS subject classification: 54A20, 54D55 In many areas of general topology, we must study the following problem: Let Y be a given space. If Y is embedded in a larger space X, how is Y located in X? It goes without saying that, when Y coincides with the larger space X, the relative property V must coin-cide with the topological property V of the space Y. The first systematic exposition of relative topological versions appeared in A. V. Arhangel'skii and H. M. M. Genedi [3] in 1989. They discussed fundamental properties like example relative Hausdorff, relative regularity and relative compactness type properites etc. and posed some open questions. Some years ago, A. V. Arhangel'skii [2] also gave valuable surveys and many important results on relative topological properties and rela-tive topological spaces. So this provides us with a guideline in our work on relative ver-sions. Some new results on relative normality and relative paracompactness were obtained by I. Ju. Gordienko [8, 9] and K. Miyazaki [11] etc. and studies of relative compactness were done in A. V. Arhangel'skii, I. V. Jaschenko and J. Tartir [4,5] etc.. On the other hand, it is known that the countable paracompactness occupies an impor-tant position among covering properties. So it will be valuable and indispensable to study the relative version of countable paracompactness. So we shall introduce the notions of relatively countable paracompactness in parallel with relative paracompactness and we shall study mutual relations among them and give their characterizations which will illuminate properties of the relative countable paracompactness. Almost results of this paper appeared in Y. Yasui [14] without proofs. Our notations and terminologies which are not defined in this paper are found in R. Engelking [7]. Key Words: general topology, relatively countable n-paracompactness, relatively locally finite, increasing cover, normality of product