proceedings of the american mathematical society Volume 9(). Number 4, April 1984 TWO RESULTS CONCERNING CARDINAL FUNCTIONS ON COMPACT SPACES I. JUHÁSZ and Z. SZENTMIKLÓSSY Abstract. We show that for X compact T2: (i) d( X) « s( X) ■F( X); (ii) if the pair (k, F( X)) is a caliber of X then w( X) < k. These strengthen results of Sapirovskii from [3 and 5], respectively. Moreover, (i) settles a problem raised in [2] implying that there are no compact T2 K-examples for any singular cardinal k. In this note we follow the notation and terminology of [1]. In particular, we let F(X) denote the smallest cardinal k such that |5|< k for any free sequence S C X. Theorem 1. If X is compact T2 then d(X) < s(X) ■ F(X). Proof. Let us put s( X) ■F( X) = k. Given any nonempty open set U E X we can choose a family Q(U) of open Fa sets in X such that U — UQ(U). But X does not contain discrete subspaces of cardinality k+ , hence, e.g. by 2.13 of [1], there is a subfamily 9>(U) E G(U) and a subset S(U) C U such that |<S(i/)|< k, \S(U)\*z k and U E U<S(I/) LIS(U). Let us now assume, indirectly, that d(X) > k. Then we also have 7r(A') > /c. Hence if & is a family of nonempty closed Gs sets in X with 16t|< k, then there is an open nonempty U E X such that A\U ¥= 0 for each A G â. It follows easily from the compactness of X that if % is a chain of open sets with this property, then U % possesses it as well. Thus by Zorn's lemma, we can fix an open set W(&) which is maximal with respect to the above property. Observe that then for every nonempty set H open in X\W(&), there is an A E & with A C H U W(&). Hence 0 # A\W(â) C H, i.e. {A\W(&): A E &} is a 77-network in X\W(&). Consequently, we have d(X\W(â)) <|6B|<k. After these preparations we define by transfinite induction, families ®a of closed Gs subsets of X for a E k with | cS>a |< k as follows. If a E k and %ß has been defined for all ß E a, we consider the open set Wa— W(U {%ß: ß E a}) and the family 9>(Wa) of open F0 subsets of Wa. For every 6 £ ^(WJ we may then choose closed Gs sets F£ for « G w such that G = L){F¿: « E <o}.<$a is then defined as the set of all nonempty finite intersections of members of the family U {<$>ß:ßEa} U [X\G: GE<$(Wa)} U [F¿: GE%(Wa),n E co}. Received by the editors January 31, 1983. 1980 Mathematics Subject Classification. Primary 54A25, 54D30. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 608 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use