The purpose of this paper is to identify the role of perfectness in the Michael insertion theorem for perfectly normal locales. We attain it by characterizing perfect locales in terms of strict insertion of two comparable lower semicontinuous and upper semicontinuous localic real functions. That characterization, when combined with the insertion theorem for normal locales, provides an improved formulation of the aforementioned pointfree form of Michael’s insertion theorem.

中文翻译：

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完善的语言环境和本地实际功能

本文的目的是确定完全正态语言环境中Michael插入定理中完美性的作用。我们通过严格插入两个可比较的下半连续和上半连续局部实函数来描述完美的语言环境来实现这一点。当与正常语言环境的插入定理结合使用时，该特征提供了迈克尔插入定理的上述无点形式的改进形式。