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1 was proved by P. Hall (1) in 1935. It is implicit in the earlier literature and has, for this reason, been sometimes associated with the names of Denes Konig and E. Egervary. 1 on bipartite graphs. However, it is precisely Hall’s formulation that has turned out to be the master key which has unlocked many closed docrs. 2. I was comparatively difficult. Other proofs of this or of closely related results have since been given by a number of writers, among them W. Maak ( l ) , Marshall Hall Jr. (2), Weyl ( l ) , Everett & Whaples (I), Halmos Vaughan (l), D.

Kelley’s book (1) contains a full discussion of this result and of other statements equivalent to the axiom of choice. Very brisk proofs of Zorn’s lemma have been given by J. D. Weston (1) and, more recently, by H . Kneser (1). 6. The standard account of general topology is to be found in Kelley (1). g. Bushaw (1). Both books (and, of course, many others) contain proofs of Tychonoffs theorem. This result is now known to be equivalent to the axiom of choice. 7. More recent accounts have been given by C .

Ex. 6). 3, the relation of inequality between cardinals is a partial order: it is also a total order, but this fact is much harder to prove. More obvious instances of total order come readily to mind. Thus any set of real numbers is totally ordered with respect to the relation of (ordinary) inequality. If ( X , <) is a partially ordered set, then we can define another partial order, say < *, on X by declaring that x < * y if and only if y d x. We shall say that < * is the reciprocal order of < .

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