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.
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Last modified February 10, 2004.