General Cardinality Restrictions
by
Dimitrina N. Stavrova
Miami University, Oxford, Ohio 45056

A theorem is obtained from which almost all known cardinality restrictions for topological and more general cases as well as some new ones follow.

Let \lambda, \tau, \mu and \nu be infinite cardinal numbers. Let X be a nonempty set and let X_o be a nonempty subset of X . Let us consider the following notions:

(1) \lambda-structure ( R.Hodel ) : For every point x of X let H(x) subset [ exp X ] ^{<= \lambda} such that \cap H(x) \owns x and let H = \cup { H(x) : x in X } . The pair ( X, H ) is called a \lambda-structure on X .

(2) If ( X, H ) is a \lambda-structure and X_o subset X we use the notation H(X_o) to denote \cup { H(x) : x in X_o } .

(3) A family L subset [X_o] ^{<= \mu} will be called \mu-dominating in X_o if for every Y in [X_o] ^{<= \mu} there is an element L in L such that Y subset L .

(4) If M is a family of subsets of X_o then Y subset X_o will be called M(\tau) -inductive if Y is an union of an increasing by inclusion subfamily of cardinality \tau of elements of M .

(5) Let N be a family of subsets of X . Then a mapping p : [ N ] ^{<= \nu} --> exp X will be called an ( N, \nu) -operator on X .

(6) Let s : [ X_o ] ^{<= \mu} --> [ X ] ^{<= \mu} be such that for every Y subset X_o with cardinality less than or equal \mu we have that Y subset s(Y) . Then s will be called a \mu-operator on X_o in X .

Theorem. Let ( X, H ) be a \lambda-structure. Let L be \lambda^\tau -dominating on X_o , let p be a ( H, \tau) -operator on X and s be a \lambda^\tau -operator on X_o in X . Let :

(*) If Y = \cup { Y_\alpha : \alpha in \tau⁺ } subset X_o is L(\tau⁺)-inductive and X_o \Y =/= \emptyset then there are - a point q in X_o an ordinal \alpha in \tau⁺ and a family \gamma in [H (s(Y_\alpha))] ^{<= \tau} such that q not in p(\gamma) and p(\gamma) contains Y .

Then | X_o | <= \lambda^\tau .

Date received: February 11, 2000