Document Type
Article
Publication Date
2018
College/Unit
Eberly College of Arts and Sciences
Department/Program/Center
Mathematics
Abstract
The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every ν ∈ ω ∪ {∞} there exists a family Fν ⊂ C ν (R) of cardinality ω1 < c such that for every g ∈ D ν (R) the set g \ S Fν has cardinality ≤ ω1. Moreover, we show that this result remains true for partial functions g (i.e., g ∈ D ν (X) for some X ⊂ R) if, and only if, ν ∈ {0, 1}. The proof of this result is based on the following theorem of independent interest (which, for ν 6= 0, seems to have been previously unnoticed): for every X ⊂ R with no isolated points, every ν-times differentiable function g : X → R admits a ν-times differentiable extension ¯g : B → R, where B ⊃ X is a Borel subset of R. The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of R, which short but fully detailed proof is included. Some open questions are also posed.
Digital Commons Citation
Ciesielski, Krzysztof, "Simultaneous Small Coverings by Smooth Functions Under the Covering Property Axiom" (2018). Faculty & Staff Scholarship. 854.
https://researchrepository.wvu.edu/faculty_publications/854