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.
