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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.)
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