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