  We introduce a new topological generalization of the $\sigma$-projective
hierarchy, not limited to Polish spaces. Earlier attempts have replaced
$^{\omega}\omega$ by $^{\kappa}\kappa$, for $\kappa$ regular uncountable, or
replaced countable by $\sigma$-discrete. Instead we close the usual
$\sigma$-projective sets under continuous images and perfect preimages together
with countable unions. The natural set-theoretic axiom to apply is
$\sigma$-projective determinacy, which follows from large cardinals. Our goal
is to generalize the known results for $K$-analytic spaces (continuous images
of perfect preimages of $^{\omega}\omega$) to these more general settings. We
have achieved some successes in the area of Selection Principles--the general
theme is that nicely defined Menger spaces are Hurewicz or even
$\sigma$-compact. The $K$-analytic results are true in ZFC; the more general
results have consistency strength of only an inaccessible.
