In this paper, well known
completeness conditions in Moore spaces are generalized to arbitrary first countable
spaces. Relationships are established between these conditions and various other
completeness concepts including Čech completeness, countable completeness, and
countable subcompactness. Finally, conditions are given for embedding a given first
countable space in a “first countable complete” space. As one application of the
theory developed, a necessary and sufficient condition is obtained for the
embedding of a Moore space in a semicomplete or “Rudin” complete Moore
space.
