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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

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

# 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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