We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such that $f^{-1}(y)$ is metrizable for all $y\in Y$ . A continuous map of compacts $f : X \to Y$ is said to be fully closed if for any disjoint closed subsets $A;B \subset X$ the intersection $f(A) \cap f(B)$ is finite. For instance the projection of the lexicographic square onto the first factor is fully closed and all its fibers are homeomorphic to the closed interval.

中文翻译：

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Fedorchuk 契约上连续函数空间的局部均匀圆整重归一化

我们证明 $C(X)$ 承认等效的逐点下半连续局部均匀圆形范数，前提是 $X$ 是谱高为 3 的 Fedorchuk 紧致。换句话说，$X$ 承认完全封闭的映射 $f$ 到度量紧致 $ Y$ 使得 $f^{-1}(y)$ 对所有 $y\in Y$ 都是可度量的。如果对于任何不相交的闭合子集 $A;B \subset X$ 的交集 $f(A) \cap f(B)$ 是有限的，则称连续映射 $f : X \to Y$ 是完全闭合的。例如，字典方格在第一个因子上的投影是完全闭合的，并且它的所有纤维都同胚于闭合区间。