4

FUNDAMENTAL CONCEPTS

antiresidual if for q(EQ and rq in S, r is in Q. The quasi-ordered set S is

directed if for any p and g in S there exists s in S satisfying sp and sq.

Such an s may be called a common successor of p and g.

One important example of a quasi-ordered set is any family S of subsets

of a given set X, ordered by inclusion O In this case a common successor of

p and q is a subset of their intersection.

For another example consider coverings of X. A covering { Un ) is called a

refinement of a covering { V^} if each Ua is a subset of at least one V^. We

write { Ua ) [V0}, and note that is a quasi-ordering.

Inclusion is actually a partial ordering, i.e., besides being transitive it is

reflexive and anti-symmetric. Note that refinement is also reflexive, but not

anti-symmetric. Two coverings °ky ^ may be equivalent in the sense that

^ ^ a n d ^ ^ .

Any two coverings {Ua\\ V0 }, of a set have a coarsest common refine-

ment, the covering { Uan V0}. It may be denoted by { Ua)A{ Vfi }. Of course,

there are usually other coverings equivalent to this one.

If^is a covering of X and A is a subset of X, the star St(A^) of A with

respect to ^is the union of all elements of °k which have a nonempty inter-

section with A, The collection [St(Uf%): U^%\ is a covering and is called

^*, the star oi°k. \i°k* is a refinement of % °k is called a star-refinement of

°y, and one writes ^ * ^ The relation * is again a quasi-ordering, gener-

ally not reflexive.

In any quasi-ordered set, a filter is a directed antiresidual subset. A filter

base is a cofinal subset of a filter, i.e., a directed set. In a partially ordered

family of sets, ordered by inclusion, a proper filter is a filter which does not

have the empty set as an element.

Now we come to the main definitions. A preuniformity /tona set X is a

family of coverings of X which forms a filter with respect to *. A uniform-

ity M on X is a preuniformity such that for any two points, x, y, of X, there is

a covering ^ in /*, no element of which contains both x and y. A uniform space

fiX is a set X with a uniformity n on X. The elements of n are called uniform

coverings.

6. A family \x of coverings is a preuniformity if and only if (i) for *% and V

in /x, ^ A ^ is in /*; (ii) for % °y and % in n V is in n; and (iii) every

element of n has a star-refinement in p.

7. For any two points, x, y, of a uniform space, there is a uniform covering

tysuch that St(x,^) and St(y, °k) are disjoint.

In some of the literature, what we call a uniform space is called a separated uniform space,

and a set with any preuniformity on it is called a uniform space. We shall use the term "sepa-

rate" mainly in the following (customary) sense: a family {/«) of functions with the same do-