### 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
- 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

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

# 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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