The notion of localic real functions was introduced by Gutiérrez García et al. with which pointfree forms of insertion theorems for some classes of topological spaces were obtained. In this paper, we present insertion theorems for countably paracompact frames and stratifiable frames in terms of localic real functions, yielding the insertion theorems for countably paracompact spaces and stratifiable spaces while applied to the frame of open subsets of these spaces.

中文翻译：

---

可数超紧致框架和可分层框架的插入定理

局部实函数的概念是由 Gutiérrez García 等人引入的。得到了一些拓扑空间类的无点形式的插入定理。在本文中，我们根据局域实函数提出了可数超紧致框架和可分层框架的插入定理，得出可数超紧空间和可分层空间的插入定理，同时应用于这些空间的开子集框架。