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By L. Mirsky

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E. 1, + t* + 1 - n + IA(1)I - 111 r* 3 n + min {IA(I)I - III}. I The assertion is therefore proved. 1. , x,}+. ’, A, (where two A’s are regarded as formally distinct even when they are equal as sets) will simply be called objects. A collection S of objects is said to be incidence-bound if the relation x i E Aj implies that at least one of x i , Aj belongs to S. Again, S is said to be incidence-free if x i $ A j whenever both x i and Aj belong to S. We shall denote by t , the minimum number of objects in an incidence-bound collection and by f the maximum number of objects in an incidence-free collection.

U A i k } and hence l{,j: I < j < pi, Bj n F # 011 = \Ai, u ... u Aikl 3 k = IF]. , B,,,); and this means that YI possesses a transversal. Finally, we mention a slight generalization of the preceding theorem. 5. , A,) be a family ofsubsets of E, and let E* 5 E. I if and only if every subset F of E* intersects at least IF1 A’s. 4. 2 1. , n} and F E E. 2. , A,,) be a family of sets which satisfies Hall’s condition. Show that the additional requirement [ A , u ... u A,,I = n is necessary and sufficient for the transversal of rU to be unique.

Let us, for the moment, call a subset of S ‘special’ if it can be written in the form X(G,: i~ I), where G i E Yifor all i and G i = S ifor all but a finite number of i. ; then Y is known as the product topology on S . The topological space (S, 9)is called the topological product of the spaces (Si,Yi), i~ I. In the course of our discussion, we shall need to invoke a result which is pivotal in the entire theory of topological spaces. 1. (Tychonoff) The topological product of a family of compact topological spaces is again compact.