### Archives

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

# 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
4 Comments