Strange Hamel bases under the Covering Property Axiom CPA
by
Krzysztof Chris Ciesielski
West Virginia University

In the talk I will present a Covering Property Axiom CPA (which is consistent with ZFC - holds in the iterated perfect set model) and discuss several of its consequences. This includes:

• There is a family H of \omega₁ pairwise disjoint perfect subsets of R such that the union of H forms a Hamel basis, that is, a linear basis of R over Q. In particular,

  • there is a non-measurable subset X of R without the Baire property which is rigid with respect of the ideal I of meager measure zero sets, that is, such that the symmetric difference of X and r+X belongs to I for every real number r,
  • there is a function f:R --> R such that for every h in R the difference function D_h(x)=f(x+h)-f(x) is Borel, but for every \alpha < \omega₁ there is an h in R such that D_h is not of Borel class \alpha.
• There exists a discontinuous, almost continuous, and additive function f:R --> R whose graph is of measure zero.
• There exists a Hamel basis H such that E⁺(H) has measure zero, where E⁺(A) is a linear combination of a subset A of R with non-negative rational coefficients.

To follow the talk only a minimal knowledge of set theory is required.

Date received: March 16, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagb-10.