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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014

### Keywords

Cohen real Diamond approachability ideal Constructible Universe Singular Density sap Forcing Axioms projective Boolean algebra diamond star 20M14 square Rado's conjecture Almost-disjoint famiy Chang's conjecture Jonsson cardinal P-Ideal Dichotomy Whitehead Problem Souslin Tree Shelah's Strong Hypothesis Non-saturation Fat stationary set HOD xbox Small forcing b-scale Parameterized proxy principle Commutative cancellative semigroups Dushnik-Miller Partition Relations Successor of Singular Cardinal Prevalent singular cardinals PFA(S)[S] very good scale Uniformization Ostaszewski square Absoluteness Cardinal function Almost Souslin Ascent Path Chromatic number Fodor-type reflection Generalized Clubs Poset Singular coﬁnality Successor of Regular Cardinal Hindman's Theorem Forcing Fast club S-Space Coherent tree middle diamond Hereditarily Lindelöf space Hedetniemi's conjecture weak square Rainbow sets Prikry-type forcing Erdos Cardinal Reduced Power incompactness super-Souslin tree Slim tree Square-Brackets Partition Relations Almost countably chromatic Erdos-Hajnal graphs square principles Martin's Axiom 05A17 Distributive tree Nonspecial tree Knaster Kurepa Hypothesis Microscopic Approach Selective Ultrafilter Club Guessing Aronszajn tree polarized partition relation Universal Sequences stationary reflection Rock n' Roll Large Cardinals Foundations Minimal Walks PFA Antichain 11P99 Luzin set coloring number Stevo Todorcevic Axiom R free Boolean algebra stationary hitting weak diamond reflection principles Postprocessing function Mandelbrot set Uniformly coherent Weakly compact cardinal ccc Cardinal Invariants Singular cardinals combinatorics L-space tensor product graph OCA Sakurai's Bell inequality

# 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 Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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