arXiv subjects
Publications and source records attributed to Joan E. Hart.
Assuming PFA, every uncountable subset E of the plane meets some C^1 arc in an uncountable set. This is not provable from MA(aleph_1), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C^2 arcs, and the counter-example is a perfect set.
We consider a class of compacta X such that the maps from X onto metric compacta define an Aronszajn tree of closed subsets of X.
Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight aleph_1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.
Assuming Jensen's principle diamond, there is a compact Hausdorff space X which is hereditarily Lindelof, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The diamond example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot contain the two earlier examples of failure of the CSWP, which were totally disconnected -- specifically, the Cantor set (W. Rudin) and beta N (Hoffman and Singer).
Let X be compact abelian group and G its dual (a discrete group). If B is an infinite subset of G, let C_B be the set of all x in X such that converges to 1. If F is a free filter on G, let D_F be the union of all the C_B for B in F. The sets C_B and D_F are subgroups of X. C_B always has Haar measure 0, while the measure of D_F depends on F. We show that there is a filter F such that D_F has measure 0 but is not contained in any C_B. This generalizes previous results for the special case where X is the circle group.
The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.
If B is an infinite subset of omega and X is a topological group, let C^X_B be the set of all x in X such that converges to 1. If F is a filter of infinite sets, let D^X_F be the union of all the C^X_B for B in F. The C^X_B and D^X_F are subgroups of X when X is abelian. In the circle group T, it is known that C^X_B always has measure 0. We show that there is a filter F such that D^T_F has measure 0 but is not contained in any C^X_B. There is another filter G such that D^X_G = T. We also describe the relationship between D^T_F and the D^X_F for arbitrary compact groups X.