On relative countable paracompactness
by
Yoshikazu Yasui
Dept. of Math.,Osaka Kyoiku Univ.

We shall 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?
The first systematic exposition of relative topological versions appeared in A.V. Arhangel'skii and H.M.M. Genedi in 1989. They discussed fundamental properties like relative Hausdorff, relative regularity and relative compactness type properites etc.. Some results on relative normality and relative paracompactness were obtained by I.Ju. Gordienko etc. and studies of relative compactness were done in A.V. Arhangel'skii, I.V. Jaschenko and J. Tartir etc..
In this talk, we shall discuss the relative version of countable paracompactness and study their characterizations etc..

Definition Let Y be a subspace of a space X.

1. Y is countably 1-paracompact in X, if every countable open cover U of X has an open cover V of X which is a refinement of U and is locally finite at Y in X, that is, every point of Y has a nbd in X which intersects with at most finitely many members of V.
2. Y is countably 2-paracompact in X, if every countable open cover U of X has a family V of open subsets of X which is a partial refinement of U and is a cover of Y and is locally finite at Y in X.
3. Y is countably 3-paracompact in X, if every countable open cover U of X has a cover V of Y, which is locally finite at Y in itself and a partial refinement of U, and whose elements are open subsets of Y.

References

1. A.V. Arhangel'skii and H.M.M. Genedi, Beginnings of the theory of relative topological properties, in: General Topology. Spaces and Mappings (MGU, Moscow, 1989) 3-48 (in Russian).
2. A.V. Arhangel'skii, Relative topological properties and relative topological spaces, Top. appli. 70(1996)
3. A.V. Arhangel'skii and I.V. Jaschenko, Relatively compact spaces and separation properties, Comm. Math. Univ. Carolin. 37(3)(1996).

Date received: June 24, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caby-18.