All the Filter Stuff in One Place and Several Things Equivalent to the Choice Axiom

Of these definitions, the first and third may make some sense, since the definition of a filter base describes the family C of sets in the “generalized Cantor intersection theorem” except for dropping the requirement that they be closed (which would not make sense in a general set X anyway). The definition of a filter is motivated in two ways. The first is that, since one is only interested in the “set-theoretically small” sets in a filter base, one can throw in the intersections and the supersets without making any difference in the “small” sets—which are the ones in which one is really interested—one can look at F(B) as just being a “maximal version of B.” What is really going on when one makes this definition, however, is the fact that a filter as thus defined is a proper dual ideal in the Boolean algebra (2 ,∪,∩). That is: consider the family J of complements in X of the elements of a given filter G. The duals (i.e., statements in terms of complements in X) of the defining properties of a filter read: