13.1. THE GENERAL CONCEPT OF A FILTER 3 CLAIM 13.2. Let J7 be a proper filter on a set X, and let Y C X. Then the family T U {Y} generates a proper filter if and only if Y is J7-stationary. Let « b e a regular uncountable cardinal, and let 7 b e a filter on a nonempty set X. We say that T is ^-complete or n-closed if p|*4 G T for every A G [^r]/t. An Ki-complete filter is also called countably complete. Every principal ultrafilter Ta is ^-complete for all K. The filter J\fx may or may not be countably complete. If it is, then x is said to be a P-point in X. The filter T\ of Example 13.2 is countably complete but not (2H°)+-complete, since the family A = {[0, l]\{x} : 0 x 1} C Tx is of cardinality 2H°, but \\A = 0 £ T\. EXERCISE 13.3 (G). (a) Show that if T is a nonprincipal ultrafilter on a set X with \X\ ft, then T is not AC+-complete. (b) Show that if an ultrafilter T on a set X is not countably complete, then there exists a denumerable family {Yn : n G LU} C T such that C\neuJ Yn = 0. If there exists a nonprincipal /^-complete ultrafilter on a set of size AC, then K is called a measurable cardinal. Measurable cardinals will be studied in Chapter 23. So far we have been talking only about filters on a set X, i.e., filters consisting of arbitrary subsets of X. But if you know some general topology, you will certainly have noticed that topologists speak about "filters of closed sets," "ultrafilters of zero sets," etc. These objects are usually not filters of the type we have discussed so far. In particular, our filters are closed under arbitrary supersets, and a superset of a closed set is not in general closed. We need a more general concept of filters. DEFINITION 13.3. Let (P, ) be a p.o., and let A CF. An element p of P is a lower bound for A if p q for every q G A. We say that p and q are compatible in A if A contains a lower bound for the set {p, q}. If p and q are compatible in P, then we just say that they are compatible and write p X Q't otherwise we say that p and q are incompatible and write p _ L q. A subset F of P is called a filter in (P, ) (or just a filter in P if the p.o. relation is implied by the context), if (I) F is closed upwards, i.e., \/p G FVg G P (p q - q G F) (II) Every finite subset of F has a lower bound in F. Note that we speak about filters in P to avoid confusion with filters on X. EXERCISE 13.4 (G). Convince yourself that if we replace (II) in the above defi- nition by the apparently weaker demand that every two elements of F are compatible in F, then we get an equivalent definition. EXAMPLE 13.4. Let P be the family of all open subsets of the unit interval of positive Lebesgue measure, and consider the p.o. (P, C). Let A(0.5) be the family of all subsets of P of measure 0.5. Then every two elements of A(0.5) are compatible in P, but not necessarily in A(0.5). Moreover, not every finite subset of A(0.5) has a lower bound in P. EXAMPLE 13.5. A filter on X is the same thing as a filter in (V{X), C), and a proper filter on X is the same as a filter in (7^(X)\{0}, C). Thus there is no need to introduce a separate notion of proper filter in a p.o. EXAMPLE 13.6. Let X be a topological space, and let K(X) = {K C X : K is closed and nonempty}. A filter F in )C(X) will be referred to as a filter of closed sets (the adjective "nonempty" is usually dropped in this context).
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