#### 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