Relative Collectionwise Normality

Authors

  • Eliser Grabner Slippery Rock University
  • Gary Grabner Slippery Rock University
  • Kazumi Miyazaki Osaka Elector-Communication University
  • Jamal Tartir Youngstown State University

DOI:

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

Keywords:

Paracompact, Collectionwise normal, Relative topological properties

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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Author Biographies

Eliser Grabner, Slippery Rock University

Dept. Math.

Gary Grabner, Slippery Rock University

Dept. Math.

Kazumi Miyazaki, Osaka Elector-Communication University

Dept. Math.

Jamal Tartir, Youngstown State University

Dept. Math. and Stat.

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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How to Cite

[1]
E. Grabner, G. Grabner, K. Miyazaki, and J. Tartir, “Relative Collectionwise Normality”, Appl. Gen. Topol., vol. 5, no. 2, pp. 199–212, Oct. 2004.

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