A characterization of completely regular spaces

This paper proves that uniform spaces and admissible spaces form the same class of topological spaces. This result characterizes a completely regular space as a topological space that admits an admissible family of open coverings. In addition, the admissible family of coverings provides an interesting methodology of studying aspects of uniformity and dynamics in completely regular spaces.

LEBESGUE COVERING LEMMA ON NONMETRIC SPACES

In this paper the Lebesgue covering lemma is extended from the setting of metric spaces to the setting of admissible spaces. An admissible space is a topological space endowed with an admissible family of open coverings, and need not be metric. The paper contains applications to uniform continuity and dimension theory.