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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

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

# 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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