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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

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

# 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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