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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

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

# Tag Archives: PFA(S)[S]

## 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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