Katetov’s problem

Abstract

Abstract. In 1948 Miroslav Katětov showed that if the cube X 3 of a compact space X satisfies the separation axiom T 5 then X must be metrizable. He asked whether X 3 can be replaced by X 2 in this metrization result. In this note we prove the consistency of this implication.

PFA(S)[S]: More Mutually Consistent Topological Consequences of PFA and V = L

“…As noted in [16], for S a coherent (König calls these uniformly coherent) Souslin tree, and s, t on the same (ηth) level of S, there is a canonical isomorphism σ S st between the cones above (we think of our trees as growing upwards) s and t, defined by letting σ S st (s )(α) be t(α) if α < η and s (α) otherwise, for each s extending s. These isomorphisms are such that σ S su = σ S tu • σ S st and σ S st = (σ S ts ) −1 . See [13] for a construction of a coherent Souslin tree from ♦.…”

“…We shall use "s above s" and "s extends s" synonymously, and admit the possibility that s = s. Before proceeding further, let us say what "coherent" means, since we will be using it. We quote from [16]; also see the references listed there, as well as [11].…”

“…The solution by Larson and Todorcevic of Katětov's problem [16] depended on showing the remarkable fact that -using the weaker c.c.c. technique -some of the "Souslin-type" consequences [12] of MA ω 1 , namely that compact, first countable, hereditarily separable spaces are hereditarily Lindelöf, and that first countable, hereditarily Lindelöf spaces are hereditarily separable, are consistent with some of the "normal implies collectionwise Hausdorff" consequences of V = L, namely that separable normal first countable spaces are collectionwise Hausdorff.…”

“…Since then, the strength of both types of consequences has been increased. Larson and Tall [14] dropped the separability in the second type of consequence, by starting with a particular ground model, while Todorcevic [26] improved [16] to get from PFA(S)[S] that compact hereditarily separable spaces are hereditarily Lindelöf. Here we obtain another result in the V = L column, getting that normal spaces which are either first countable or locally compact are collectionwise Hausdorff.…”

Club-Isomorphisms of Aronszajn Trees in the Extension with a Suslin Tree

“…This is one of the consequences of }. Larson and Todorčević [13], [14] demonstrated some consequences of } which hold in the extension with a Suslin tree. These consequences are implied by }.R; ¤/, which is also forced with a Suslin tree.…”