In [23], Todorcevic gives a survey of basis problems in combinatorial set theory, listing nine theorems and six working conjectures — all in the presence of PFA — including the following three of interest to us here: Conjecture 1 (Todorcevic; [23]). If R is a binary relation, then either R ≤ א0 ·ω1 or [ω1]0 ≤ R. Conjecture 2 (Hajnal, Juhasz; [5]). If X is a regular Hausdorff space, then the fol...

In this paper we study the L-convergence of the Riesz means S R(f) for the sublaplacian on the sphere S in the complex n-dimensional space C. We show that S R(f) converges to f in L (S) when δ > δ(p) := (2n− 1)| 2 − 1 p |. The index δ(p) improves the one found by Alexopoulos and Lohoué [AL], 2n| 1 2 − 1 p |, and it coincides with the one found by Mauceri [Ma] and, with different methods, by Mül...

We construct, assuming Jensen’s principle ♦, a one-dimensional locally connected hereditarily separable continuum without convergent sequences.

We show that hereditarily indecomposable spaces can be characterized by a special instance of the Intermediate Value Theorem in their ring of continuous functions.

A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace. (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X .) The H.I. spaces form a new and, as we believe, fundamental class of Banach spaces. The celebrated example of a Banach space with no...

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.