UNIFORMITIES AND PREUNIFORMITIES

3

Another important point: the metric uniform space has more structure

than the topological space. There may be two homeomorphic metric uniform

spaces which are not uniformly equivalent; and more.

5. There is an uncountable family of countable discrete metric spaces, no two

of which are uniformly equivalent to each other.

PROOF.

From topology we know that there is an uncountable family of

compact subspaces of the plane, no two of which are homeomorphic with

each other.

(Recall the construction. For any increasing sequence of positive integers,

nin2 • • • begin with the segment from (0,0) to (0,1) in the plane and at

each point (0, 2"') attach nt short whiskers.)

Let {Ca | be such a family of spaces in the plane. In each Ca select a counta-

ble dense subset

\pan\.

Let the coordinates of

pan

be (x?n,yn). Let Xa be

the subset of three-space consisting of all points (x£, y£, 1/m), with m^n.

Then Xa is a countable metric space. It is discrete since each of its points is

above the horizontal coordinate plane and for each e 0 there are only finitely

many points of Xa with third coordinate greater than «. Of course, Xa is not

closed in three-space; its closure Ya consists of Xa and a copy of Ca. More-

over, y« is the completion of Xa. Now suppose /: Xa -^X0 is a uniform equiv-

alence. In particular, / maps Xa uniformly continuously into the complete

space Y0. Hence / has a unique uniformly continuous extension g: Ya—Yp.

The same reasoning applies

torl:X0—Xa,

which must have a uniformly

continuous extension h: Y(,—Ya.

Finally, consider the composed mapping hg of Ya into itself. On the dense

set Xa, hg coincides with the identity mapping i. But for any two continuous

mappingsp :A—Btq: A—£, where A and B are Hausdorff spaces, the set

of all a in A such that p(a) =^q(a) is a closed set. Hence hg: Ya —Ya is the

identity. Similarly gh: Y0-+Yp is the identity. But then g maps Ca homeo-

morphically onto C0f which is absurd. It follows that Xa and X0 cannot be

uniformly equivalent.

This result illustrates a useful rule of thumb in the theory of uniform

spaces: All counterexamples are discrete.

Uniformities and preuniformities. We shall need a little of the terminology

of the theory of quasi-ordered sets. Moreover, it will be convenient to use

slightly nonstandard terminology; so every reader should note carefully the

following definitions.

A set S is said to be quasi-ordered by a relation if is transitive. A sub-

set Q of S is cofinal in S if for each element s of S there exists an element q of

Q such that7 s. Q is residual in S if whenever qGQ and r q in S, r£Q. Q is