In this paper, we give some characterizations of $k$-semistratifiable and $k$-MCM by expansions of set-valued mappings. It is shown that for a space $X$, the following statements are equivalent: (1) $X$ is $k$-semistratifiable; (2) for every metric space $Y$, there exists an order-preserving operator $\Phi$ that assigns each set-valued mapping $\varphi: X \rightarrow \mathcal {F}(Y)$ ($\mathcal {F}(Y)$ is the set of all nonempty closed set of $Y$), a l.s.c. and $k$-u.s.c. set-valued mapping $\Phi(\varphi): X \rightarrow \mathcal {F}(Y)$ such that $\Phi(\varphi)(x)$ is bounded for each $x\in U_\varphi$, where $U_\varphi=\{x\in X: \varphi \text{~is locally bounded at ~}x\}$, and that $\varphi\subseteq \Phi(\varphi)$.
