### 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

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

# 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
4 Comments