  The purpose of this article is to define and study new invariants of
topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These
invariants take values in the $p$-adic numbers and are constructed from a
virtual pro-$p$ completion of the fundamental group. The key result of the
article is an approximation theorem which shows that the $p$-adic invariants
are limits of their classical analogues. This is reminiscent of L\"uck's
approximation theorem for $L^2$-Betti numbers.
  After an investigation of basic properties and examples we discuss the
$p$-adic analog of the Atiyah conjecture: When do the $p$-adic Betti numbers
take integer values? We establish this property for a class of spaces and
discuss applications to cohomology growth.
