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

  • YIN-ZHU GAO (a1) and WEI-XUE SHI (a2)
Abstract

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.

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Copyright
Corresponding author
For correspondence; e-mail: yzgao@nju.edu.cn
Footnotes
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This project is supported by NSFC (No. 10971092).

Footnotes
References
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[1]Bennett, H. R. and Lutzer, D. J., ‘Problems in perfect ordered spaces’, in: Open Problems in Topology, (eds. van Mill, J. and Reed, G. M.) (Elsevier Science, Amsterdam, 1990), pp. 231236.
[2]Bennett, H. R. and Lutzer, D. J., ‘Continuous separating families in ordered spaces and strong base conditions’, Topology Appl. 119 (2002), 305314.
[3]Gao, Y.-Z. and Shi, W.-X., ‘The D-property and the Sorgenfrey line’, Bull. Aust. Math. Soc. 80 (2009), 233236.
[4]Halbeisen, L. and Hungerbühler, N., ‘On continuously Urysohn and strongly separating spaces’, Topology Appl. 118 (2002), 329335.
[5]Miwa, T. and Kemoto, N., ‘Linearly ordered extensions of GO-spaces’, Topology Appl. 54 (1993), 133140.
[6]Popvassilev, S. G. and Porter, J. E., ‘Monotonically D-spaces’, Topology Proc. 30 (2006), 355365.
[7]Shi, W.-X., ‘Perfect GO-spaces which have a perfect linearly ordered extension’, Topology Appl. 81 (1997), 2333.
[8]Shi, W.-X. and Gao, Y.-Z., ‘Sorgenfrey line and continuous separating families’, Topology Appl. 142 (2004), 8994.
[9]Shi, W.-X., Miwa, T. and Gao, Y.-Z., ‘A perfect GO-space which cannot densely embed in any perfect orderable space’, Topology Appl. 66 (1995), 241249.
[10]Stepanova, E. N., ‘Extension of continuous functions and metrizability of paracompact p-spaces’, Math. Notes (Mat. Zametki) 53 (1993), 308314.
[11]van Douwen, E. K. and Pfeffer, W. F., ‘Some properties of the Sorgenfrey line and related spaces’, Pacific J. Math. 81 (1979), 371377.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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