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Realcompactification of frames. (English) Zbl 0840.54027
The author modifies a construction of G. M. Schlitt [$$\mathbb{N}$$-Compact frames and applications, Doctoral thesis, McMaster University, 1990; see also Commentat. Math. Univ. Carol. 32, No. 1, 173-187 (1991; Zbl 0747.06009)] to construct realcompactifications of a completely regular frame $$L$$ by forming the appropriate quotient of the frame envelope of a regular sigma-frame which join generates $$L$$. If $$A$$ is taken as the sigma-frame of all cozero elements of $$L$$ then the corresponding realcompactification is the universal one, the realcompact coreflection in the category of frames of $$L$$. One motivation for the definition of realcompactness for frames used here is that a topological space is realcompact if and only if its topology is realcompact as a frame. Note then [J. Madden and J. Vermeer, Math. Proc. Camb. Philos. Soc. 99, 473-480 (1986; Zbl 0603.54021)] the analogues of well-known characterizations of realcompactness for spaces give rise to distinct concepts in the setting of frames.

##### MSC:
 54D60 Realcompactness and realcompactification 18B99 Special categories 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54J05 Nonstandard topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
##### Keywords:
regular sigma-frame; realcompactification; realcompactness
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