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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
- 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

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

# 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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