One of the reasons
that paracompactness plays a central role in general topology is that it is a
property shared by compact spaces and metric spaces. Recently there has
been considerable interest in topological properties shared by countably
compact spaces and metric spaces. R. W. Heath has introduced a method of
describing a generalized metric property of a topological space (X,τ) by means of
a function g : N × X → τ and R. E. Hodel has modified this method to
obtain important new classes of spaces. Subsequently, J. Nagata obtained a
similar characterization of Σ#-spaces, and it now appears that the method
of Heath and Hodel provides an opportunity to clarify the relationships
among those properties that are shared by countably compact spaces and
metric spaces. This note seeks to establish some relationships among these
properties.
