  A class of topological spaces is projective (resp., $\omega$-projective) if
and only if projective systems of spaces (resp., with a countable cofinal
subset of indices) in the class are still in the class. A certain number of
classes of Hausdorff spaces are known to be, or not to be, ($\omega$-)
projective. We examine classes of spaces that are not necessarily Hausdorff.
Sober and compact sober spaces form projective classes, but most classes of
locally compact spaces are not even $\omega$-projective. Guided by the fact
that the stably compact spaces are exactly the locally compact, strongly sober
spaces, and that the strongly sober spaces are exactly the sober, coherent,
compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we
examine which classes defined by combinations of those properties are
projective. Notably, we find that coherent sober spaces, compact coherent sober
spaces, as well as (locally) strongly sober spaces, form projective classes.
