need help with b and c

(b) Let U : A > 'J'(P) be a function such that, for every oz 6 A, Ua is an open subset of P. Show that nUa 0:64 is open. (In words: You are showing that the intersection of open subsets is open.) (0) Let P and Q be posets. A function f : P > Q will be called continuous if and only if the preiinage of any open subset of Q is an open subset of P. In other words, if V C Q is open, then f'1(V) is open in P. Prove that f is continuous if and only if it is a map of posets. Notation Let 8 be a collection of some sets. (In many of our examples, 5 will be the power set of some set.) Let A be a set. And x a function S : A > 8. Out of laziness, if or E .11, instead of writing S (a), we will write 80, for the value of S on 0:. Various authors use different notation for this idea: (a) Instead of writing out 8 as a function, they may write {5.1564, i.e., a collection of sets indexed by a. (b) They may simply say, \"choose a set 80, for each a E .11.\" Using the \"a set is a bag with stuff in it\" language, the data of {Sahe is simply a chosen array of bags. How many bags are there? There are as many bags as there are elements in A. Note that some bags may be repeatedS may be noninjective. Warning 0.0.1. Keep in mind that A may be an innitely large set! Then U 8., (0.0.0.1) 056A is the union of all the sets 80,. In other words, we take all the bags and dump their contents into a single large bag. This single large bag is written Uae Sm as in (0.0.0.1). Be warned that some bags may contain \"the same elements,\" and these elements are not doublecounted, or doubledup, in the big bag. This is just the usual notion of union of sets. Likewise, 0 3a 05604 is the intersection of all the sets SQ. In other words, we look through all the bags, and determine if there are elements that are contained in every bag in sight. We form a new bag, ag, SQ, consisting only of those elements that appear in every bag