集合自蒂茨性质

AI总结：

本文引入集合自蒂茨性质，研究上半连续集值函数的扩展问题，证明紧致度量空间具有该性质，而环面不满足自蒂茨性质。

摘要：

我们引入了集合自蒂茨性质，这是上半连续集值函数的自蒂茨性质的类比。一个拓扑空间$X$是自蒂茨的，如果对于每一个闭集$A \subseteq X$和连续函数$f \colon A \to X$，存在一个连续扩展$F \colon X \to X$扩展$f$。一个拓扑空间$X$是集合自蒂茨的，如果对于每一个闭集$A \subseteq X$和上半连续集值函数$f \colon A \to 2^X$，存在一个上半连续集值函数$F \colon X \to 2^X$使得$\left. F \right|_A = f$。我们证明每个紧致度量空间都是集合自蒂茨的，而环面不满足自蒂茨性质。

英文摘要：

We introduce the set-self-Tietze property, an analogue of the self-Tietze property for upper semi-continuous set-valued functions. A topological space $X$ is self-Tietze, if for every closed $A \subseteq X$ and continuous function $f \colon A \to X$, there is a continuous extension $F \colon X \to X$ of $f$. A topological space $X$ is set-self-Tietze, if for every closed $A \subseteq X$ and upper semi-continuous set-valued function $f \colon A \to 2^X$, there exists an upper semi-continuous set-valued function $F \colon X \to 2^X$ such that $\left. F \right|_A = f$. We show every compact metric space is set-self-Tietze, and that the torus is not self-Tietze.

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