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Distributive tree Rock n' Roll middle diamond tensor product graph PFA(S)[S] Aronszajn tree Sakurai's Bell inequality Respecting tree Selective Ultrafilter GMA Souslin Tree stick Uniformly homogeneous Prevalent singular cardinals Non-saturation polarized partition relation Intersection model Dushnik-Miller Almost-disjoint family 54G20 Sigma-Prikry HOD Cardinal Invariants Constructible Universe Rado's conjecture Erdos-Hajnal graphs Filter reflection Closed coloring Strongly Luzin set Diamond for trees Successor of Singular Cardinal stationary reflection Knaster S-Space weak Kurepa tree Large Cardinals higher Baire space Was Ulam right Strongly compact cardinal Microscopic Approach Greatly Mahlo Generalized descriptive set theory full tree very good scale Amenable C-sequence xbox Commutative projection system Square-Brackets Partition Relations transformations Fodor-type reflection Prikry-type forcing Countryman line b-scale unbounded function Luzin set Shelah's Strong Hypothesis Kurepa Hypothesis PFA club_AD Uniformly coherent Weakly compact cardinal Strong coloring Commutative cancellative semigroups free Boolean algebra Lipschitz reduction Cohen real Knaster and friends Precaliber L-space Generalized Clubs free Souslin tree Absoluteness Cardinal function weak diamond Open Access Antichain Axiom R P-Ideal Dichotomy approachability ideal specializable Souslin tree Small forcing Erdos Cardinal Singular Density Hedetniemi's conjecture Foundations ccc Almost Souslin indecomposable ultrafilter Rainbow sets Minimal Walks Postprocessing function Ramsey theory over partitions Subtle cardinal strongly bounded groups countably metacompact Diamond positive partition relation Forcing Singular cofinality Iterated forcing incompactness Parameterized proxy principle Well-behaved magma Martin's Axiom Slim tree Ulam matrix C-sequence Successor of Regular Cardinal Local Club Condensation. Sierpinski's onto mapping principle Subnormal ideal Subtle tree property Hindman's Theorem Ineffable cardinal reflection principles AIM forcing Reflecting stationary set Uniformization regressive Souslin tree sap Subadditive Poset Fat stationary set nonmeager set coloring number Chromatic number diamond star Ascent Path Ostaszewski square Reduced Power stationary hitting square principles Jonsson cardinal ZFC construction projective Boolean algebra Coherent tree Nonspecial tree OCA Diamond-sharp Partition Relations Vanishing levels O-space weak square super-Souslin tree Mandelbrot set Analytic sets square Singular cardinals combinatorics Hereditarily Lindelöf space Fast club Dowker space Chang's conjecture Universal Sequences Forcing Axioms Club Guessing Whitehead Problem Almost countably chromatic SNR
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
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