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

# 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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