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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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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