Sobolev embedding theorems and their generalizations for maps defined on topological spaces with measures

For mappings from measure space $(X,\mu)$ to Banach space $(Y,|\cdot|_Y)$ we defined an analogous of Sobolev classes $W_p^r(X;Y)$, $r=1,2,\dots$, and also Sobolev–Slobodetsky classes $W_p^r$, $r\in [1,\infty)$, and some of their generalizations. We prove the embedding theorems into $L_q$ and into Orlizc classes and study some properties of Sobolev functions.