Small coverings with smooth functions under the Covering Property Axiom

by

Krzysztof Ciesielski, and J. Pawlikowski

Canad. J. Math. 57(3) (2005), 471-493.

In the paper we formulate a Covering Property Axiom CPA, which holds in the iterated perfect set model, and show that it implies the following facts.

• There exists a family F of less then continuum many C1 functions from R to R such that R2 is covered by functions from F in the sense that for every (x,y) in R2 there exists an f in F such that either f(x)=y or f(y)=x.
• For every Borel function f:R-->R there exists a family F of less than continuum many "C1" functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f.
• For every positive n and a Dn function f:R-->R there exists a family F of less than continuum many Cn functions whose graphs cover the graph of f.
We also provide the examples showing that in the above properties the smotheness conditions are the best possible.