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The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A $$\pi $$ π -compactification of a frame L is a compactification constructed using a special type of a basis called a $$\pi $$ π -compact basis; the Freudenthal compactification is the largest $$\pi $$ π -compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.
The N-star compactifications of frames are the frame-theoretic counterpart of the N-point compactifications of locally compact Hausdorff spaces. A $$\pi $$ π -compactification of a frame L is a compactification constructed using a special type of a basis called a $$\pi $$ π -compact basis; the Freudenthal compactification is the largest $$\pi $$ π -compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all N-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which N-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.
We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales S with the property that the ambient frame has a basis, elements of which induce open sublocales whose frontiers miss S. This notion is stronger than the traditional zero-dimensionality of a sublocale. A compactification of a frame is perfect if its associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect compactifications of frames is introduced, and we show that it contains all the perfect ones. Indeed, not every rim-perfect compactification is perfect, but compactifications with a ZDE remainder do not distinguish between rim-perfectness and perfectness. The Freudenthal compactification has a ZDE remainder. We show that a frame L is rim-compact if and only if L has a compactification with a ZDE remainder. Several results concerning perfect compactifications and ZDE remainders are provided.
In this article, algebraic characterisations of J-spaces and C-normal spaces are exhibited. The concept of a Z-connected ideal in C(X) is presented and characterised using certain connected subsets of X. We define the class of JC-spaces and characterise its members via Z-connected ideals. Two more classes of ideals in C(X), namely the coz-free and F-free ideals, are instituted. These types of ideals are used to establish conditions under which a given space is a strong J-space. We introduce the notion of a J-lattice and show that the lattice, CL(X), of closed subsets of X is a J-lattice if and only if X is a J-space. A pointfree topology exposition of J-lattices is also presented, with more attention to complete Boolean algebras.
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