Joan E. Hart

Single axioms for odd exponent groups

With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human.

Bohr topologies and compact function spaces

Abstract We consider (discrete) structures, A , for a countable language. A # denotes A with its Bohr topology. Let Y be a compact Hausdorff space. Then Y is homeomorphic to a subspace of some A # iff Y is Talagrand compact.

On the regularity of the topology of separate continuity

Abstract Given topological spaces X,Y, there is a unique topology T + on X×Y such that, for all topological spaces Z, a function f :X×Y→Z is continuous with respect to T + iff f is separately continuous. We consider situations under which T + is regular or normal. This is related to Eberlein compacta in the case that X,Y are compact, and to σ-sets in the case that X,Y are separable metric.

First Countable Continua and Proper Forcing

Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight ℵ1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Weak measure extension axioms

Abstract We consider axioms asserting that Lebesgue measure on the real line may be extended to measure a few new nonmeasurable sets. Strong versions of such axioms, such as real-valued measurability, involve large cardinals, but weak versions do not. We discuss weak versions which are sufficient to prove various combinatorial results, such as the nonexistence of Ramsey ultrafilters, the existence of ccc spaces whose product is not ccc, and the existence of S - and L -spaces. We also prove an absoluteness theorem stating that assuming our axiom, every sentence of an appropriate logical form which is forced to be true in the random real extension of the universe is in fact already true.