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Abstract.Let X be a Tx topological space. Let a(X) = sup{a: X has a closed discrete subspace of cardinality a} and v(X) = min{a: A^ can be written as the intersection of a open subsets of X x X); here Ay denotes the diagonal {(x,x): x e X] oi X. It is proved that \X\ < exp^X") v(X)). If, in addition, X is Hausdorff, then X has no more than expia^) viX)) compact subsets.1. Introduction. There are several known relationships among the cardinal functions on a topological space that involve the cardinalities of closed discrete subsets of the space. Among these are Jones's lemma (see, for example, [10, p. 100]) and the recent results of Burke and Hodel [2]. It is the purpose of this note to add to this list of relationships.The authors wish to thank the referee for several useful suggestions and, in particular, for pointing out that Theorem 2.6 is a consequence of Theorem 2.1.Information about cardinal functions on topological spaces appears in Juhasz [6]; we shall use the notation and terminology of this text. We shall henceforth assume that all hypothesized topological spaces are Tx. Any additional separation axioms used in the proof of a theorem will be set forth explicitly in the statement of the theorem. The cardinality of a set X is denoted by | A'|; [A'] will denote the set of two-element subsets of X. Cardinal numbers are identified with the set of ordinals preceding them. The smallest cardinal greater than the cardinal X is denoted by X+. We shall use the following settheoretic theorem, due to Erdös and Rado; see [4].

We give a unified treatment of several fixed-point type theorems by using the concept of dismantlability, extended from ordered sets to arbitrary graphs. For a graph G and a vertex x of G we let NG(X) denote the set of neighbours of x in G. We say that x is a subdominant vertex of G if there is a vertex y of G, distinct from x, such that NG(x)∪{x} ⊆ NG(y)∪{y}. If G has n vertices we say that G is dismantlable if the vertices of G can be listed as x1, x2, ..., xi,..., xn such that, for all i = 1,2,..., n— 1, xi is a subdominant vertex of the graph Gi = G — {xj : j < i}