2 13. FILTERS AND IDEALS IN PARTIAL ORDERS EXERCISE 13.1 (G). Let J7 be a, proper filter on a nonempty set X. (a) Show that if X J7*, then X satisfies the following conditions: (i)* X is closed under subsets, i.e., V 7 G I V Z C F ( Z G T) (ii)* X is closed under finite unions, i.e., |J H G X for all H G [J]a . (b) Show that if a family X C P(X) satisfies canditions (i)* and (ii)*, then the family X* = {Y C X : X\Y G X} is a filter on X. (c) Show that F** = T. (d) Show that a set Y C X is ^"-stationary if and only if Y D Z ^ 0 for all A family J C 'P(X) that satisfies conditions (i)* and (ii)* of Exercise 13.1(a) is called an ideal on X. An ideal X on X is proper if X ^ V(X), i.e., if X £ X. If X is a proper ideal on X, then Z* is a proper filter on X. The family of X*-stationary sets is often denoted by X+. One can think of a proper filter on X as a family of "large" subsets of X the dual ideal would then be the family of "small" subsets of X, and the stationary sets would be "not too small" in a sense. Example 13.2 nicely illustrates this point of view: T{ consists of all subsets of the unit interval of measure zero, whereas the T\-stationary sets are precisely the sets of positive outer measure. In Example 13.3, a subset Y of X is A4-stationary if and only if x is in the closure of Y. In Example 13.1, the family of .^-stationary sets is Ta itself. The reason is that Ta is an ultrafilter. In our new terminology, Theorem 9.10 can be expressed as follows: CLAIM 13.1. Let T be a proper filter on a nonempty set X. Then T is an ultrafilter if and only if J7 coincides with the family of J7-stationary sets. In the proof of Theorem 9.10, we constructed a filter that contained a given subfamily of V(X). Let us now review this construction. For A C V(X), let flt(A) = {YCX:3ke u3A0,... , Ak^ G A (Y D A0 H n Afc_i)}. In particular, if J7 is a filter on X and Y C X, then the set flt(F,Y) defined in Chapter 9 is the same as fit (J7 U {Y}) in our new terminology. We say that a subfamily A of V(X) has the finite intersection property (abbreviated fip) if Ao fl fl Ak 7^ 0 for every finite subset {A0,... , Ak} of A. We call A a filter base if for every finite subset {Ao,... ,Ak} of A there exists A G A such that A c A0n••• n Ak. EXERCISE 13.2 (G). Let X ^ 0 and A C V(X). (a) Show that the family flt(A) is the smallest filter on X that contains A. (b) Show that fit (A) is proper iff A has the fip. (c) Show that flt(A) = {Y C X : 3A G A (A C Y)} iff A is a filter base. (d) Show that if X is infinite, then every uniform filter on X is contained in a uniform ultrafilter on X. We call flt(A) the filter generated by A. Note that if Nx is as in Example 13.3 and B C V(X), then B is a filter base with flt(B) = Mx if and only if B is a neighborhood base at x. Lemma 9.11(b) says that if T is a filter on X and Y C X, then flt(TU {Y}) is proper if and only if X\ Y ^ .F. This translates into the following characterization of F-stationary sets:
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