Notes on forcing axioms (Q2866031)

The content of the book is divided into 17 chapters and 3 appendices. The author assumes that the reader is acquainted with forcing and the topology background of basic forcing axioms. Two appendices at the end of the book additionally explain several notions that are assumed to be commonly known. The third appendix contains historical remarks. The following notes come from the content of the chapters:NEWLINENEWLINE1.~The Baire category numbers for several classes of topological spaces are considered: the class of separable completely metrizable spaces, the class of separable Čech-complete spaces, the class of ccc Čech-complete spaces, the class of Čech-complete spaces for which the Baire category number is not provably~\(\omega_1\). These cardinals are known to be, respectively, the covering of category, \(\mathfrak{p}\), \(\mathfrak{m}\), and the cardinal \(\mathfrak{mm}\) connected with Martin's maximum. The cardinal \(\mathfrak{mm}\)~is connected with the class of forcing notions preserving stationary subsets of~\(\omega_1\). Some basic relations between this class and any of the classes of ccc, \(\sigma\)-closed, proper, and semi-proper forcing notions are proved.NEWLINENEWLINE2.~Subsets of sets of cardinality~\(<\mathfrak{p}\) can be coded by subsets of~\(\mathbb{N}\) and the existence of such coding has an influence on the cardinal arithmetic and the structure of closed nowhere dense sets. Also codings of subsets of \([\mathfrak{p}]^2\) and subsets of \(\mathfrak{p}\times\mathfrak{p}\) by sequences of subsets of~\(\mathbb{N}\) of length~\(\mathfrak{p}\) are presented.NEWLINENEWLINE3.~Any bijection between subsets of cardinality~\(<\mathfrak{p}\) of two Polish spaces can be extended to a~Borel isomorphism. Assuming \(\mathfrak{p}>\omega_1\) the existence of an uncountable co-analytic set without a~perfect subset has several characterizations.NEWLINENEWLINE4.~The additivity of a~locally finite Radon measure is~\(\geq\mathfrak{m}\) and if \(\mathfrak{m}>\omega_1\), then every locally finite outer-regular Radon measure space is \(\sigma\)-finite.NEWLINENEWLINE5.~Every ccc poset of cardinality~\(<\mathfrak{m}\) is \(\sigma\)-centered and consequently, if \(\mathfrak{m}>\omega_1\), the Souslin hypothesis holds. If \(\mathfrak{m}>\omega_1\), then applying coherent mappings the existence of a~selective ultrafilter on~\(\omega\) which is \(\Sigma_1\)-definable in \((H_{\omega_1},\in)\) is proved. If \(\mathfrak{m}>\omega_1\), then every compact countably tight ccc space and every compact \(T_5\) ccc space is separable.NEWLINENEWLINE6.~Some results concerning the \(S\)-space problem and the \(L\)-space problem under the assumption \(\mathfrak{m}>\omega_1\) are presented.NEWLINENEWLINE7.~The open graph axiom (OGA) is the statement saying that for every open graph on~a~separable metric space~\(X\) either \(G\)~has an uncountable complete subgraph or \(G\)~is countably chromatic. If \(\mathfrak{mm}>\omega_1\), then OGA holds.NEWLINENEWLINE8.~The small ideal dichotomy concerning \(\omega_1\)-generated ideals, the sparse set-{\hskip0pt}mapping principle, and the \(P\)-ideal dichotomy are consequences of \(\mathfrak{mm}>\omega_1\).NEWLINENEWLINE9.~Lipschitz trees and coherent trees and ultrafilters defined from coherent trees are investigated under the assumption \(\mathfrak{m}>\omega_1\). Assuming \(\mathfrak{mm}>\omega_1\) every coherent tree has a~certain coloring property.NEWLINENEWLINE10.~The assumption \(\mathfrak{mm}>\omega_1\) positively solves the \(S\)-space problem and produces a~solution of the von Neumann problem. Hence, every complete weakly distributive Boolean algebra supports a~positive continuous submeasure.NEWLINENEWLINE11.~Every nonseparable Banach space of density~\(<\mathfrak{mm}\) admits a~quotient space with a~monotone Schauder basis of length~\(\omega_1\) and consequently it contains an uncountable biorthogonal system. If \(\mathfrak{mm}>\omega_1\), then a~Banach space is separable if and only if every closed convex subset contains a~point which does not support it. This positively answers a~problem of Rolewicz. If \(\mathfrak{m}>\omega_1\), then a~compact space~\(K\) is metrizable if and only if all biorthogonal systems in \(C(K)\) are countable.NEWLINENEWLINE12.~If \(\mathfrak{mm}>\omega_1\), then every compact countably tight space is sequential and every compact ccc \(T_5\)~space is Fréchet. Also ultrapowers of \(\omega\) are not embedable into the reduced powers and every norm on a~Banach algebra of the form \(C(K)\) is equivalent to the supremum norm.NEWLINENEWLINE13.~If \(\mathfrak{mm}>\omega_1\), then \(\omega_2\not\to(\omega_2,\omega+2)\) and \(\omega_1\to(\omega_1,\alpha)\) for all \(\alpha<\omega_1\).NEWLINENEWLINE14.~If \(\mathfrak{mm}>\omega_1\), then \(1\), \(\omega\), \(\omega_1\), \(\omega\times\omega_1\), and \([\omega_1]^{<\omega}\) are the only cofinal types of directed sets of cardinality at most~\(\aleph_1\).NEWLINENEWLINE15.~If \(\mathfrak{mm}>\omega_1\), then every set of reals of size~\(\aleph_1\) embeds into any uncountable separable linear ordering and there is a~list of five orderings~\(\Sigma\) such that a~linearly ordered set is countable if and only if it contains no isomorphic copy of any ordering from~\(\Sigma\).NEWLINENEWLINE16.~If \(\mathfrak{mm}>\omega_1\), then \(\mathfrak{mm}=\mathfrak{c}=\omega_2\) and \(\theta^{\aleph_1}=\theta\) for all regular \(\theta\geq\omega_2\).NEWLINENEWLINE17.~If~\(\mathfrak{mm}>\omega_1\), then the strong reflection principle \(\text{SRP}(\lambda)\) holds for all cardinals~\(\lambda\), i.e., for every \(S\in[\lambda]^\omega\) and \(a\in H_{\lambda^+}\) there is a~continuous \(\in\)-chain of countable elementary models \(N_\xi\prec H_{\lambda^+}\) for \(\xi<\omega_1\) containing~\(a\) such that \(N_\xi\cap\lambda\in S\) if and only if there is \(M\prec H_{\lambda^+}\) such that \(N_\xi\subseteq M\), \(M\cap\omega_1=N_\xi\cap\omega_1\), and \(M\cap\lambda\in S\). Also some weaker reflection principles are considered.