UNIFORMITIES AND PREUNIFORMITIES

5

mainX but possibly different ranges separates points provided x^y in X implies that for some

a,fa{x)?*fa(y). Other uses of the word are introduced in places, particularly in Chapter VI.

The uniformities or preuniformities on any set form a partially ordered set

under inclusion. The preuniformities, like the topologies, form a complete

lattice. (This is a corollary of Proposition 9 below.) Evidently, any preuni-

formity which contains a uniformity is a uniformity. As with topologies, a

preuniformity M containing a preuniformity v is said to be finer than v. The

usage of the terms "strong" and "weak" is not standardized, and we shall

avoid them as far as possible.

The weak uniformity induced by a family of functions is too useful to be

avoided, and fortunately there has been little or no terminological confusion

here. We have

8.

THEOREM.

For any family \ fa\ of functions on a set X into various uni-

form spaces, there is a coarsest preuniformity on X including all the inverse

images of uniform coverings under these functions. If the functions separate

points, then this preuniformity is a uniformity.

This uniformity is the weak uniformity induced by the family \fa). The

proof of Theorem 8 is not difficult, but we shall take some time marking out

important ideas in it.

A basis for a uniformity n is a filter base for n considered as a filter of cov-

erings; and similarly for a preuniformity. A sub-basis for a uniformity or

preuniformity is a family of coverings whose finite intersections form a basis.

Now a family v of coverings which satisfies condition (iii) of Proposition 6,

every covering in v has a starrefinement in v, is called a normal family. It is

convenient and customary to use the term normal sequence for something

more special than a sequence which is a normal family: specifically, for a

sequence of coverings

%n

such tha*

%n+l* %n

for each n.

9. Every normal family of coverings is a sub-basis for a preuniformity.

PROOF.

The required preuniformity is the family of all coverings which

can be refined by finite intersections of coverings from the given family,

which automatically satisfies (i) and (ii) of 1.6. For (iii) we need only observe

that if

9^i*^i

for i = l , ...,n,then ^

1

A - - - A ^

, I

* ^

l

A - - - A ^

n-

PROOF OF THEOREM

8. The operation /; on coverings preserves star-

refinements; so the inverse images of uniform coverings form a normal fam-

ily. Then this is a sub-basis for a preuniformity /*, which is the coarsest

possible.

We should note that a sub-basis need not be a normal family; of course, a

basis must.