Vol. 33, No. 2, 1970

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Mesocompactness and related properties

Vincent Mancuso

Vol. 33 (1970), No. 2, 345–355

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.

Mathematical Subject Classification
Primary: 54.50
Received: 18 August 1969
Published: 1 May 1970
Vincent Mancuso