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This paper is concerned with
some of those generalizations of paracompactness which can arise by broadening the
concept of local finiteness, e.g., metacompactness, in contrast to those which come
about by varying the power of an open cover, e.g., countable paracompactness. Quite
recently, several generalizations of the first type have been studied. These include
mesocompactness and sequential mesocompactness, strong and weak cover
compactness, and Property Q.
In §1, the notion of metacompactness (= pointwise paracompactness) is
used to establish a hierarchy among these concepts, and in regular γ-spaces,
some of these notions are shown to be equivalent to paracompactness. In §2,
it is shown that mesocompactness is an invariant, in both directions, of
perfect maps and that unlike paracompact spaces, there exists a mesocompact
T3 space which is not normal, and a mesocompact T2 space which is not
regular.
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