Vol. 44, No. 1, 1973

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A note on mesocompact and sequentially mesocompact spaces

James Robert Boone

Vol. 44 (1973), No. 1, 69–74

Characterizations of paracompactness in k-spaces have been obtained which employ the notions of mesocompact and sequentially mesocompact spaces. Property (ω) has been used to characterize collectionwise normality in sequential spaces, and applied to the study of metrizability of developable spaces. It is the purpose of this paper present mapping theorems, in §3, which establish the invariance properties of normal mesocompact spaces, under perfect mappings, and normal sequentially mesocompact spaces, under closed presequential mappings. For this purpose, characterizations of these structures are developed in §2, and the notion of a presequential mapping is introduced in §3. These characterizations are obtained by the use of two generalizations of collectionwise normality, property (k) and property (ω), which are defined and studied in §1. Characterizations of collectionwise normality and paracompactness in spaces with property (k) are presented in §4.

Mathematical Subject Classification 2000
Primary: 54D20
Secondary: 54C10
Received: 14 September 1971
Published: 1 January 1973
James Robert Boone