Kurt Gödel Research Center

2016 seminar talk: A forcing built around a coherent Souslin tree and its uses for normal, locally compact spaces

Talk held by Peter Nyikos (University of South Carolina, Columbia, USA) at the KGRC seminar on 2016-06-28.

Abstract

A form of forcing involving ground models with coherent Souslin trees was invented by Paul Larsen and Stevo Todorcevic in order to lay to rest a 1948 problem of Katetov, who had shown that a compact space $X$ is metrizable iff either $X^2$ is perfectly normal or $X^3$ is hereditarily normal (abbreviated $T_5$: this means every subspace is normal).

In 1977 I found a nice example if there is a Q-set, of a space $X$ where $X^2$ is $T_5$ but $X$ is not metrizable, and later Gary Gruenhage found a completely different example under CH. Larson and Todorcevic found a model in 2002 where there are no such examples.

Their technique consisted of forcing from a ground model with a coherent Souslin tree $S$ to get all ccc posets $P$ that presere $S$ to have filters meeting any collection of $< \mathfrak c$ dense subsets of $P$ [Such models are referred to by the shorthand MA(S).] and then forcing with $S$ itself, resulting in MA(S)[S] models.

These models have "paradoxical" properties, satisfying some consequences of V=L such as "every first countable normal space is collectionwise Hausdorff" and some consequences of MA($\omega_1)$ such as "every separable locally compact normal space is hereditarily separable and hereditarily Lindelöf."

Since then, the technique has been expanded to replace "ccc" with "proper" to give PFA(S)[S] models and very recently to replace it with "semi-proper" to give MM(S)[S] models. Locally compact spaces of various sorts have been shown to have a host of simplifying properties in these models. One striking recent example:

Theorem. In MM(S)[S] models, every locally compact, $T_5$ space is either hereditarily paracompact or contains a copy of the ordinal space $\omega_1$.

Many other examples will be surveyed and shown not to follow just from ZFC. For instance, a Souslin tree with the order topology is a (consistent!) counterexample to the topological statement in the preceding theorem.