2
FUNDAMENTAL CONCEPTS
PROOF.
Suppose /is uniformly continuous and ^ i s a covering of Y having
Lebesgue number c. If 5 is such that any two points of X at distance less than
5 have images at distance less than «, then the collection of (5/2)-neighbor-
hoods of all points of X is the required uniform covering^.
Conversely, suppose the condition on coverings is satisfied and e 0. L e t ^
be the covering of Y consisting of all sets of diameter *, and ^ a uniform
covering of X each element of which has its image contained in a single ele-
ment of^ If 6 is a Lebesgue number for^then any two points of X at dis-
tance 6 have images at distance «.
A uniformly continuous function / : X— Y is called a uniform equivalence if
/is one-to-one and onto and the inverse function
f~l:Y-X
is also uniformly
continuous.
3. If f:X-+Y is a uniform equivalence, then a collection [Ua] of subsets of
Xisa uniform covering if and only if the collection {/(£/„)} is a uniform cover-
ing of Y. The converse is also true.
The proof is left as an exercise.
Finally, we may define a metric uniform space as a set X together with a
family /* of coverings of X such that for at least one distance function d on X,
M is precisely the family of all uniform coverings of the metric space (X, d).
The preceding remarks and results show that every metric space determines
uniquely a metric uniform space; that two different distance functions d, e,
on the same set X, determine the same uniform space if and only if the iden-
tity mapping is a uniform equivalence between (X, d) and (X, e); and further,
if we are given the metric uniform spaces (X,/i) and (Y,v) and a function
/ : X—*Y, we can determine whether / is uniformly continuous by 1.2, without
knowing or constructing any specific distance functions.
This summary treatment of metric uniform spaces will not be used in
developing the general theory. It is included in justification; we propose to
define such terms as "uniformly continuous" and "completion \ and we
ought to show that the notions are true generalizations of the familiar no-
tions for metric spaces. The remaining details in this showing will be swept
into exercises or omitted.
Another important point is that, while we lose the distance function in
passing from metric space to metric uniform space, we do not lose the topol-
ogy. For example,
4. For metric spaces X and Y, a function /: X— Y is continuous if and only
if for each point x in X, for each uniform covering ^of Y, there exist an ele-
ment V of V and a uniform covering % of X such that for every element U of
°k which contains x, /(!/) C V".
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