Some of the notions of the theory of uniform spaces are familiar from
metric spaces. It is possible to found the entire theory on the notion of a set
with a family of pseudometrics called a gage. We shall not do this here, but
we shall lead in gradually from metric notions to the idea which will be the
foundation of the development: the uniform coverings.
Metric uniform spaces. Recall that a pseudometric d on a set X is a real-
valued function on X X X satisfying d(x, y) =d(y, x) ^ 0 and d(x, z)+d(z, y) ^
d(xty), for all a, y, z in X. It is called a metric if it separates points, i.e., x^y
implies d(x,y) 0. A metric space consists of a set X with a metric d on X.
Commonly we refer to "the metric space X" and use the letter d freely for
the distance in any metric space.
Recall that a function /: X—• Y, where X and Y are metric spaces, is called
uniformly continuous if for each €0 there is 50 such that whenever
d(x, x') 6 in X, d(f(x), f(x')) c. Every uniformly continuous function is con-
tinuous, but the converse is not true.
A covering ^rof X is called a uniform covering provided there is a positive
number e such that every subset of X of diameter less than e is a subset of
some element of %. Such an * is called a Lebesgue number for °k.
1. A covering % of a metric space X is uniform if and only if there is 80
such that for each point x in X, the b-neighborhood of x is contained in some ele-
ment of ^.
If^is uniform with Lebesgue number *, then every ^^-neigh-
borhood of a point is contained in an element of*2fc Conversely, if elements of
% contain all 5-neighborhoods then 8 is a Lebesgue number for ^C
2. A function /:X-~Y is uniformly continuous if and only if for every uni-
form covering V of Y there is a uniform covering °kof X such that, for each
element Uof% f(U) is contained in some element ofW.