Under what may be thought
of as a guise of a description of pathology are indicated here certain ways in which
Çech completeness, Arhangel’skii’s p-space concept, and metacompactness
enlarge on the respective concepts of metric absolute Gδ’s, metrizability, and
paracompactness. This is done through examination of certain aspects of the theory
of multivalued mappings. It is taken as a point of orientation that the topic of
Tychonoff locally bicompact spaces has a substantial mathematical interest. It is
assumed obvious that such spaces are locally paracompact p-spaces. An
underlying point of view is that the class of regular locally paracompact
p-spaces extends along natural lines the class of regular locally metrizable
spaces.
