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\(\alpha\)-paracompact subsets and well-situated subsets. (English) Zbl 0664.54015
In this paper \(\alpha\)-paracompact and well-situated subsets are further examined. A subset E of a space X is \(\alpha\)-paracompact if every covering of E by open sets has a refinement by open sets, locally finite in X, which covers E [C. E. Aull, Proc. 2nd Prague Topol. Symp. 1966, 45-51 (1967; Zbl 0162.264)] and is well-situated in X if for every paracompact \(T_ 2\) space Y, \(E\times Y\) is \(\alpha\)-paracompact in \(X\times Y\) [H. W. Martin, Topology Appl. 12, 305-313 (1981; Zbl 0483.54011)]. Covering properties of \(\alpha\)-paracompact and well- situated subsets are obtained, \(\alpha\)-paracompact and well-situated subsets are characterized in regular spaces, the behavior of \(\alpha\)- paracompact and well-situated subsets under perfect mappings is studied, and it is shown that the class of all paracompact \(T_ 2\) spaces which are well-situated in every paracompact \(T_ 2\) space in which they are embedded as closed subsets, is perfect.
Reviewer: Ch.Dorsett

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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References:
[1] C. E. Aull: Paracompact subsets. Proc. Second Prague. Topological Symposium (1966) 45-51.
[2] R. Engelking: General Topology. Polish Scientific Publishers, Warszawa, 1977. · Zbl 0373.54002
[3] H. W. Martin: Linearly ordered covers, normality and paracompactness. Top. and its Appl. 12 (1981) 305-313. · Zbl 0483.54011
[4] R. Telgársky: \(C\)-scattered and paracompact spaces. Fund. Math. 73 (1971) 59-74. · Zbl 0226.54018
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