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 \(X\) is perfectly normal or \(X\) 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 \(X\) 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.