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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.)
##### Keywords:
$$\alpha$$-paracompact subsets; well-situated subsets
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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 [5] R. Telgársky: Concerning product of paracompact spaces. Fund. Math. 74 (1972) 153-159. · Zbl 0231.54018
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