Covering Property Axiom CPA_{cube} and its consequences
by
Krzysztof Ciesielski,
and J. Pawlikowski
Fund. Math. 176(1) (2003), 6375.
In the paper we formulate a Covering Property Axiom CPA_{cube},
which holds in the iterated perfect set model,
and show that it implies easily the following facts.

For every subset S of R of cardinality continuum there exists a uniformly
continuous function g:R>R with g[S]=[0,1].
 If a subset S of R is either perfectly meager or universally null
then S has cardinality less than continuum.
 The cofinality of the measure ideal is \omega_{1}.
 For every uniformly bounded sequence
f_{n}:R>R of Borel
functions there are the sequences:
{P_{\xi}:\xi<\omega_{1}} of compact subsets of R
and
{W_{\xi}:\xi<\omega_{1}} of infinite subsets of \omega
such that the sets P_{\xi} cover R and for every
\xi<\omega_{1}:
{f_{n}P_{\xi}: n in W_{\xi}} is a monotone uniformly
convergent sequence of uniformly continuous functions.
 Total failure of Martin's Axiom:
\continuum>\omega_{1} and
for every nontrivial ccc forcing P there
exists \omega_{1}many dense sets in P such
that no filter intersects all of them.
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Last modified June 4, 2003.