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There are two main aims of the paper. The first one is to extend thecriterion for the precompactness of sets in Banach function spaces to thesetting of quasi-Banach function spaces. The second one is to extend thecriterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbbR^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach functionspaces. These criteria are applied to establish compact embeddings of abstractBesov spaces into quasi-Banach function spaces. The results are illustrated onembeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq\infty$, into Lorentz-type spaces.

Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces