k-半层空间的集值映射扩张

Expansions of set-valued mappings on k-semistratifiable spaces

摘要:

为了刻画k-半层空间引进k-半连续集值映射的定义，通过集值映射扩张刻画了k-半层空间和k-MCM空间. 住要证明了：对于空间X下列论断等价：（1）X是k-半层空间；（2）对每个度量空间Y，存在保序算子$\Phi$使得对每个集值映射$\varphi: X \rightarrow \mathcal {F}(Y)$都对应下半连续和k-上半连续集值映射$\Phi(\varphi): X \rightarrow \mathcal {F}(Y)$使得 $\Phi(\varphi)(x)$ 在每个点$x\in U_\varphi$有界并且$\varphi\subseteq \Phi(\varphi)$.

Abstract:

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 $\varphi\subseteq \Phi(\varphi)$.