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.
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