  We study a strengthening of the notion of a universally meager set and its
dual counterpart that strengthens the notion of a universally null set.
  We say that a subset $A$ of a perfect Polish space $X$ is countably perfectly
meager (respectively, countably perfectly null) in $X$, if for every perfect
Polish topology $\tau$ on $X$, giving the original Borel structure of $X$, $A$
is covered by an $F_\sigma$-set $F$ in $X$ with the original Polish topology
such that $F$ is meager with respect to $\tau$ (respectively, for every finite,
non-atomic, Borel measure $\mu$ on $X$, $A$ is covered by an $F_\sigma$-set $F$
in $X$ with $\mu(F)=0$).
  We prove that if $2^{\aleph_0}\leq \aleph_2$, then there exists a universally
meager set in $2^{\mathbb N}$ which is not countably perfectly meager in
$2^{\mathbb N}$ (respectively, a universally null set in $2^{\mathbb N}$ which
is not countably perfectly null in $2^{\mathbb N}$).
