Another result analogous to Birkhoff's representation theorem, but applying to a broader class of lattices, is the theorem of that any finite join-distributive lattice may be represented as an antimatroid, a family of sets closed under unions but in which closure under intersections has been replaced by the property that each nonempty set has a removable element.
32.
In general we must select, for each element of the index set, an element of the nonempty set of limits of the projected ultrafilter base, and of course this uses AC . However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using ( BPI ), and in fact the converse also holds.
33.
More generally, if X is a nonempty set and \ lambda is a cardinal, then C \ subseteq [ X ] ^ \ lambda is " club " if every union of a subset of C is in C and every subset of X of cardinality less than \ lambda is contained in some element of C ( see stationary set ).
34.
In other words, a nonempty set equipped with the proximal relator \ mathcal { R } _ { \ delta _ { \ Phi, \ varepsilon } } has underlying structure provided by the proximal relator \ mathcal { R } _ { \ delta _ { \ Phi } } and provides a basis for the study of tolerance near sets in X that are near within some tolerance.
35.
If the method is applied to an infinite sequence ( " X " " i " : " i " " ? ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
36.
Each choice function on a collection " X " of nonempty sets is an element of the family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all " distinct " sets in the family.
37.
From this bit vector viewpoint, a concrete Boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length ( more generally, indexed by the same set ) and closed under the bit vector operations of bitwise'", ( ", and ? as in 1010'" 0110 = 0010, 1010 ( " 0110 = 1110, and ?010 = 0101, the bit vector realizations of intersection, union, and complement respectively.
38.
The result is an explicit choice function : a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on . ( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number " k ", every family of " k " nonempty sets has a choice function . " ) This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
39.
The result is an explicit choice function : a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on . ( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number " k ", every family of " k " nonempty sets has a choice function . " ) This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
