On the cardinality of open n-in-finite families
by
S. A. Peregudov
State University of Management

All spaces are assumed to be T₁ . Pseudocompact and p-spaces are assumed to be completely regular. If \tau and \lambda are cardinals,  then one says that a family \Cal B of sets is \tau- in - <= \lambda if for every set A of cardinality \tau,  A subset B holds for no more than \lambda members B in \Cal B . \tau- in - < \lambda is similarly defined. One says that a family is \tau - in - finite in place of \tau- in - < \omega. Symbol n denotes a natural number (n >= 1).

Theorem 1 (MA + not CH). If X is a space with s(X) = \omega,  then for every open n-in-finite family \Cal B ,   |\Cal B | <= \omega (it is known in ZFC under n=1).

Theorem 2. The cardinality of each open T₁-separating points of infinite order n-in-finite family in a Baire p-space is no more than the cellularity of the space.

Theorem 3. The cardinality of each open T₁-separating points of infinite order n-in-finite family in a pseudocompact space is no more than the cellularity of the space.

The condition of separation in Theorems 2 and 3 can be omitted under n=2. Besides,  the conclusion of Theorem 2 is known for Baire spaces under n=1 with no plumages . The same is true for Theorem 3.

Define p_n (X)=sup {|\Cal S |\colon \Cal S is an open n-in-finite family in X }. 

Theorem 4. Let \Cal S be an open n-in-finite cover of a space X and let \alpha be a cardinal such that p_n (S) <= \alpha for every S in \Cal S . Then there exists an open n-in-finite refinement \Cal G of \Cal S such that for each G^' in \Cal G ,   |{G in \Cal G \colon G \cap G^' =/= \emptyset}| <= \alpha. 

Date received: June 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-26.