Hisahiro Tamano asked the
following question that motivated his Some properties of the Stone-Cech
compactification, J. Math. Soc. Japan, vol. 12, 1 (1960) 104–117. “What is a
necessary and sufficient condition for a uniform space to have a paracompact
completion?”. Although he did not arrive at a solution, he obtained characterizations
of completeness, paracompactness, and the structure of the completion of a uniform
space by means of the radical.
This paper provides an answer to the above question and related ones. Also
included is a development of other types of completeness in both uniform and
non-uniform topological spaces. These other “complete” spaces yield characterizations
of paracompactness and the Lindelöf property as well as necessary and sufficient
conditions for paracompact spaces to be Lindelöf and entirely normal spaces to be
paracompact.
