Chapter 51: Cardinal functions
511
Definitions
Cardinal functions of partially ordered sets, topological spaces,
Boolean algebras and measures; precalibers; ideals of sets.
512
Galois-Tukey connections
Supported relations; Galois-Tukey connections; covering numbers,
additivity, saturation, linking numbers; simple products; sequential
composition of supported relations.
513
Partially ordered sets
Saturation and the Erdös-Tarski theorem; cofinalities of cardinal
functions; Tukey functions; Tukey equivalence of directed sets;
σ-additivities; *metrizably compactly based directed sets;
*measurable Tukey functions.
514
Boolean algebras
Stone spaces; cardinal functions of Boolean algebras;
order-preserving functions of Boolean algebras; regular open
algebras; regular open algebras of partially ordered sets;
finite-support products.
515
The Balcar-Franek theorem
Boolean-independent sets; free subalgebras; refining systems; the
0Balcar-Franek theorem; the Pierce-Koppelberg theorem; regular
open algebras of powers of {0,1}.
516
Precalibers
Precalibers of supported relations; and Galois-Tukey connections;
partially ordered sets, topological spaces and Boolean algebras;
saturation and linking numbers; saturation of product spaces.
517
Martin numbers
Characterizations of m(P); regular open algebras, Stone
spaces and Novák numbers; precalibers, saturation and weak
distributivity; m, mcountable, p and
mK.
518
Freese-Nation numbers
Freese-Nation numbers of partially ordered sets; Boolean algebras;
upper and lower bounds for FN(A) under special axioms;
tight filtrations and Geschke systems; large algebras are not
tightly filtered.
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