\(\alpha\)-paracompact subsets and well-situated subsets.

*(English)*Zbl 0664.54015In 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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\textit{F. G. Lupiáñez}, Czech. Math. J. 38(113), No. 2, 191--197 (1988; Zbl 0664.54015)

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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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