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  • Richard Ball
Richard Ball
University of Denver

Corresponding Author:[email protected]

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Abstract

It is a celebrated theorem of Isbell that every locale \(L\) has a smallest dense sublocale, namely its complete Boolean algebra \(\mathfrak{B}L\) of pseudocomplements, aka its booleanization. In this note we prove two completely regular companion results to Isbell’s Theorem: every completely regular locale \(L\) has a smallest dense \(C^{*}\)-embedded (\(C\)-embedded) sublocale \(\mathfrak{T}L\) (\(\mathfrak{T}^{*}L\)). The following is false. Immediate consequences include the fact that \(L\) has no proper dense \(C^{*}\)-embedded (\(C\)-embedded) sublocale iff it is the sublattice (sub-\(\sigma\)-frame) sublocale generated by \(\mathfrak{B}L\), and the fact that every dense sublocale of \(L\) is \(C^{*}\)-embedded (\(C\)-embedded) iff \(\mathfrak{B}L\) is a sublattice (sub-\(\sigma\)-frame) of \(L\).