  The purpose of this paper is to introduce $\omega$-Chebyshev-greedy and
$\omega$-partially greedy approximation classes and to study their relation
with $\omega$-approximation spaces, where the latter are a generalization of
the classical approximation spaces. The relation gives us sufficient conditions
of when certain continuous embeddings imply different greedy-type properties.
Along the way, we generalize a result by P. Wojtaszczyk as well as characterize
semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous
result by the first author.
