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### Recent blog posts

- 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
- Partitioning the club guessing January 22, 2014

### Keywords

Slim tree OCA Fast club Cardinal function Forcing Axioms Prikry-type forcing xbox square Poset Martin's Axiom Universal Sequences Hindman's Theorem Minimal Walks Rado's conjecture Singular Density Singular cardinals combinatorics b-scale Hereditarily Lindelöf space Non-saturation S-Space Cohen real Square-Brackets Partition Relations Successor of Regular Cardinal free Boolean algebra PFA(S)[S] approachability ideal Souslin Tree L-space Partition Relations Selective Ultrafilter Weakly compact cardinal Hedetniemi's conjecture Uniformization Singular Cofinality Jonsson cardinal Knaster Rock n' Roll 20M14 coloring number Whitehead Problem Prevalent singular cardinals Antichain weak square Diamond tensor product graph Cardinal Invariants Almost-disjoint famiy weak diamond Singular coﬁnality Commutative cancellative semigroups 05D10 Microscopic Approach Aronszajn tree Ostaszewski square stationary reflection ccc Stevo Todorcevic Generalized Clubs Small forcing P-Ideal Dichotomy Club Guessing Constructible Universe Erdos Cardinal Reduced Power Mandelbrot set Absoluteness reflection principles Chromatic number Dushnik-Miller Almost Souslin sap Erdos-Hajnal graphs Successor of Singular Cardinal Large Cardinals Forcing Sakurai's Bell inequality Shelah's Strong Hypothesis very good scale Foundations Rainbow sets 11P99 stationary hitting Fat stationary set 05A17 Axiom R middle diamond Coherent tree polarized partition relation Parameterized proxy principle HOD incompactness Kurepa Hypothesis projective Boolean algebra square principles diamond star Almost countably chromatic PFA Fodor-type reflection Chang's conjecture Ascent Path

# 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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