A chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space. (Sample questions: (a) How large a fmily of pairwise disjoint open sets does the space admit? (b) From an uncountable family of open sets, can one always extract an uncountable subfamily with the finite intersection property. This monograph, which is partly fresh research and partly expository (in the sense that the authors co-ordinate and unify disparate results obtained in several different countries over a period of several decades) is devoted to the systematic use of infinitary combinatorial methods in topology to obtain results concerning chain conditions. The combinatorial tools developed by P. Erdos and the Hungarian school, by Erdos and Rado in the 1960s and by the Soviet mathematician Shanin in the 1940s, are adequate to handle many natural questions concerning chain conditions in product spaces.

An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. From the first (logical) property arises its connection with two-valued logic and model theory; from the second (convergence) property arises its connection with topology and set theory. Both these descriptions of an ultrafilter are connected with compactness. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic); and the convergence property leads to the process of completion by the adjunction of an ideal element for every ultrafilter-i. e., to the Stone-Cech com pactification process (implying the Tychonoff theorem on the compact ness of products). Since these are two ways of describing the same mathematical object, it is reasonable to expect that a study of ultrafilters from these points of view will yield results and methods which can be fruitfully crossbred. This unifying aspect is indeed what we have attempted to emphasize in the present work."

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