© 1996, Topology Atlas
Uniform spaces are in between metric spaces and topological spaces. A uniform space is metrizable iff it admits a countable uniform base. A topological space is uniformizable iff it is completely regular. The uniform spaces were introduced in 1937 by Andre Weil in polemic against the massive use of countability properties as a tool unifying metric spaces and topological groups. The uniform setting is the most natural one for the concepts of uniform continuity and uniform convergence. Strict analogies with metric structures are distinctly visible in some results and constructions, but uniform structures have a range of applicability considerably larger than the metric ones.
Uniform categorical aspect has been intensively investigated by the Cezch School and with uniformity in hyperspaces and in function spaces is an actual research object.
There are in literature three different approaches to uniformity. Now only convenience discriminates them. The original Weil's formulation was repalced by Bourbaki's diagonal version, the orthodox one, as a collection of neighborhoods or "entourages" of the diagonal; and by Tukey's covering version, the heretical one, appeared in 1940, adopted later in 1964 by Isbell in a survey-monograph, as a collection of coverings which is a filter with respect to the star-refinement relation. In 1976 Gillmann and Jerison in developing a theory of rings of continuous functions decided to use esclusively the description as a collection of pseudometrics or "gage". The diagonal and gage descriptions appear as more natural generalizations of the metric structures but the covering approach allows to develop a satisfactory dimension theory for uniform spaces by using combinatorial methods.
In fifties a great impetus came to the uniform theory from the Shirota result: supposing measurable cardinals don't exist, a Tyconoff space has a compatible complete uniformity iff it can be embedded as a closed subspace in a product of copies of the reals; and from the publication of Efremovic's paper on proximal geometry of metric spaces. Uniformity plays a central role in extension theories as the Stone-Cech compactification, the Hewitt real-compactification and the Tamano-Morita paracompactification.