Relative Collectionwise Normality

Eliser Grabner

United States

Slippery Rock University

Dept. Math.

Gary Grabner

United States

Slippery Rock University

Dept. Math.

Kazumi Miyazaki

Japan

Osaka Elector-Communication University

Dept. Math.

Jamal Tartir

United States

Youngstown State University

Dept. Math. and Stat.

Submitted: 2013-11-28

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Accepted: 2013-11-28

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DOI: https://doi.org/10.4995/agt.2004.1970
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Keywords:

Paracompact, Collectionwise normal, Relative topological properties

Supporting agencies:

This research was not funded

Abstract:

In this paper we study properties of relative collectionwise normality type based on relative properties of normality type introduced by Arhangel’skii and Genedi.

Theorem Suppose Y is strongly regular in the space X. If Y is paracompact in X then Y is collectionwise normal in X.

Example A T2 space X having a subspace which is 1− paracompact in X but not collectionwise normal in X.

Theorem Suppose that Y is s- regular in the space X. If Y is metacompact in X and strongly collectionwise normal in X then Y is paracompact in X.

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References:

A. Arhangel’skii, “From classic topological invariants to relative topological properties”, Scientiae Math. Japonicae 55 No 1 (2002), 153-201.

A. Arhangel’skii and H Genedi, “Beginnings of the theory of relative topological properties”, Gen. Top. Spaces and Mappings (MGU, Moscow, 1989), 3-48 (in Russian).

A. Arhangel’skii and H Genedi, “Position of subspaces in spaces: relative versions of compactness, Lindel¨of properties, and separation axioms”, Vestnik Moskovskogo Universiteta, Mathematika 44 No. 6 (1989), 67-69.

A. Arhangel’skii and I. Gordienko, “Relative symmetrizability and metrizability”, Comment. Math. Univ. Carol. 37 No 4 (1996), 757-774.

R. Engelking, “General Topology” (PWN, Warsaw, 1977).

I. Gordienko, “On Relative Properties of Paracompactness and Normality Type”, Moscow Univ. Nath. Bul. 46 No. 1 (1991), 31-32.

E. Grabner, G. Grabner and K. Miyazaki, “ Properties of relative metacompactness and paracompactness type”, Topology Proc. 25 (2000), 145-178.

K. Miyazaki, “On relative paracompactness and characterizations of spaces by relative topological properties”, Math. Japonica 50 (1999), 17-23.

J.C. Smith and L.L. Krajewski, “Expandability and collectionwise normality”, Trans. Amer. Math. Soc. 160 (1971), 437-451.

Y. Yasui, “Results on relatively countably paracompact spaces”, Q and A in Gen. Top. 17 (1999), 165-174.

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