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 \omega₁.
• For every uniformly bounded sequence f_n:R-->R of Borel functions there are the sequences: {P_\xi:\xi<\omega₁} of compact subsets of R and {W_\xi:\xi<\omega₁} of infinite subsets of \omega such that the sets P_\xi cover R and for every \xi<\omega₁: {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>\omega₁ and for every non-trivial ccc forcing P there exists \omega₁-many dense sets in P such that no filter intersects all of them.

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Last modified June 4, 2003.