Handbook of set-theoretic topology by Kenneth Kunen, K. Kunen, J. Vaughan

By Kenneth Kunen, K. Kunen, J. Vaughan

This instruction manual is an creation to set-theoretic topology for college kids within the box and for researchers in different parts for whom ends up in set-theoretic topology should be appropriate. the purpose of the editors has been to make it as self-contained as attainable with no repeating fabric that can simply be present in normal texts. The guide includes specific proofs of center effects, and references to the literature for peripheral effects the place area used to be inadequate. incorporated are many open difficulties of present interest.

In common, the articles will be learn in any order. In a number of situations they happen in pairs, with the 1st one giving an common therapy of a subject matter and the second extra complicated effects. those pairs are: Hodel and Juhász on cardinal capabilities; Roitman and Abraham-Todorčević on S- and L-spaces; Weiss and Baumgartner on models of Martins axiom; and Vaughan and Stephenson on compactness houses.

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Example text

Moreover, one can conclude from the two theorems proved below that (1) c(X) ,;::; 2w ; (2) X is a ccc space if each finite product of {X, : t E T} is a ccc space. Nevertheless, the question cannot be settled in ZFC. KuREPA [1950] has proved that if X is a Souslin line (a linearly ordered ccc space not separable), then X x X is not a ccc space. On the other hand, if one assumes MA(w1), then the product of any finite number (and hence any number) of ccc spaces is ccc. (See KuNEN [1980], p. 6. THEOREM.

Note that H has no limit points. The maximal property of H implies that D = U peH[st(p, "W') n D]. Since IDI = K and lst(p, W) n DI ::s;; r for each p E H, the set H must have cardinality K. (b) There is a sequence (xn) in Y such that I V n DI > An whenever V is any open neighborhood of Xn. First assume that (xn) has no cluster points. Then there is a subsequence (xn) of (xn) and a discrete open collection { V; : i < w} such that Xn; E V; for all i < w. Since (V, n D) is a discrete subset of Y cardinality >An; � A;, there exists B; � ( V; n D) such that IBd > A; and B; has no limit points.

Since X is an infinite Hausdorff space, the number of regular open sets in X is infinite. Consequently, it suffices to show that I C(X)I ,,;;; I RO(X)l w . Reason : RO(X) is a complete Boolean algebra, and Pierce has proved that IBl w = IBI for any infinite complete Boolean algebra B. For f E C(X), let f* denote the function from Q (the set of rational numbers) into RO(X) defined by f*(r) = (f- 1 ((-oo, r)t)0 • Define

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