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-
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