  We study spaces that can be mapped onto the Baire space (i.e. the countable
power of the countable discrete space) by a continuous quasi-open bijection. We
give a characterization of such spaces in terms of Souslin schemes and call
these spaces $\pi$-spaces. We show that every space that has a Lusin $\pi$-base
is a $\pi$-space and that every second-countable $\pi$-space has a Lusin
$\pi$-base. The main result of this paper is a characterization of continuous
open images of $\pi$-space.
