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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

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

# 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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