ABSTRACT. We say that a family A C [A]K is strongly almost disjoint if something more than just \A n B\ < n, e.g. that \A f! B\ < a < k, is assumed for A, B G A. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each A E A there is F(A) G [A]

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Abstract. We show that for every cardinal k > <•> and an arbitrary topological space AT if we have w(Y) < k whenever Y c X and | Y\ < k then w(X) < k as well. M. G. Tkacenko proved this for T3 spaces in [2]. We also prove an analogous statement for the «--weight if k is regular.The main aim of this paper is to prove the following result.Theorem. Let X be an arbitrary topological space and k > a a (regular) cardinal. If w'Y) < k (it(Y) < k) holds whenever Y c X and \Y\ < k then w(X) < k (ir(X) w be a regular cardinal. Moreover let (Ya: a E k) be an increasing sequence of subspaces of X (i.e. Ya c Yß if a < ß). If § is a family of open subsets of X such that G \ Ya is a base (tr-base) Sor Ya for each a E k, then G [ Y is also a base (ir-base) for Y = U { Ya: a G k) provided that X contains no ¡eft-separated subspace of cardinality k (or equivalents, every subspace of X has a dense subset of cardinality less than k, cf. [1]).Proof. We shall give the proof for the case of a base only, since that of the ir-base is completely analogous. Suppose, on the contrary, that § \ Y is not a base for Y.

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