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A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

Part of: Special properties Fairly general properties Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  07 February 2011

YIN-ZHU GAO  and
WEI-XUE SHI
YIN-ZHU GAO*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China (email: yzgao@nju.edu.cn)
WEI-XUE SHI
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China (email: wxshi@nju.edu.cn)
*
✉For correspondence; e-mail: yzgao@nju.edu.cn
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Abstract

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For any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and be the minimal dense linearly ordered extension of X. The following results are obtained.

  1. (1) The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.

  2. (2) The projection mapping , ϕ(〈x,i〉)=x, is closed.

  3. (3) X* is a monotone D-space if and only if X is a monotone D-space.

  4. (4) is a monotone D-space if and only if Xu is a monotone D-space.

  5. (5) For the Michael line M, is a paracompact p-space, but not continuously Urysohn.

Keywords

paracompactnesscontinuously Urysohncompactnessmonotone Dgeneralized ordered spaces

MSC classification

Secondary: 54C10: Special maps on topological spaces (open, closed, perfect, etc.) 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.) 54D30: Compactness 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 463 - 469
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This project is supported by NSFC (No. 10971092).

References

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