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Aronszajn tree Iterated forcing Ulam matrix full tree polarized partition relation Weakly compact cardinal Closed coloring Hereditarily Lindelöf space sap b-scale Ramsey theory over partitions 54G20 Dushnik-Miller Coherent tree xbox Fat stationary set Luzin set Ostaszewski square Chang's conjecture Strongly Luzin set Lipschitz reduction club_AD AIM forcing Dowker space Singular cofinality Strong coloring nonmeager set specializable Souslin tree Fast club Uniformization Parameterized proxy principle square principles ZFC construction Was Ulam right Analytic sets Cardinal function diamond star Diamond Rainbow sets positive partition relation Sakurai's Bell inequality very good scale free Souslin tree O-space Subtle cardinal Almost countably chromatic Absoluteness Selective Ultrafilter unbounded function Subadditive weak diamond Square-Brackets Partition Relations regressive Souslin tree Club Guessing Forcing Axioms OCA Martin's Axiom Universal Sequences Successor of Regular Cardinal Reduced Power Almost-disjoint family Non-saturation super-Souslin tree GMA Hindman's Theorem Cohen real Prikry-type forcing indecomposable ultrafilter Postprocessing function Subnormal ideal P-Ideal Dichotomy Erdos Cardinal Nonspecial tree Reflecting stationary set Prevalent singular cardinals Sigma-Prikry Subtle tree property Axiom R approachability ideal Greatly Mahlo Sierpinski's onto mapping principle Almost Souslin Uniformly coherent PFA Commutative cancellative semigroups higher Baire space Mandelbrot set ccc Singular cardinals combinatorics weak square C-sequence tensor product graph transformations Souslin Tree Filter reflection Singular Density Partition Relations free Boolean algebra Knaster stationary reflection projective Boolean algebra Open Access Constructible Universe Uniformly homogeneous square Forcing stationary hitting Distributive tree Precaliber Vanishing levels Fodor-type reflection Kurepa Hypothesis Diamond-sharp Rado's conjecture SNR Rock n' Roll Small forcing Diamond for trees Jonsson cardinal Large Cardinals coloring number middle diamond Ineffable cardinal Cardinal Invariants Amenable C-sequence Microscopic Approach Chromatic number Shelah's Strong Hypothesis stick L-space Foundations PFA(S)[S] HOD Generalized Clubs countably metacompact Well-behaved magma Whitehead Problem Hedetniemi's conjecture Slim tree Successor of Singular Cardinal Generalized descriptive set theory Antichain Local Club Condensation. incompactness S-Space Minimal Walks Knaster and friends Poset strongly bounded groups Ascent Path Erdos-Hajnal graphs reflection principles
Tag Archives: S-Space
A guessing principle from a Souslin tree, with applications to topology
Joint work with Roy Shalev. Abstract. We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out … Continue reading
Posted in Publications, Souslin Hypothesis, Topology
Tagged club_AD, Dowker space, O-space, regressive Souslin tree, S-Space, Souslin Tree, Vanishing levels
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Syndetic colorings with applications to S and L
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
An $S$-space from a Cohen real
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let $\mathbb C=({}^{<\omega}\omega,\subseteq)$ be the notion of … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
4 Comments