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