Article
MSC:
06B23,
18A40,
54B30,
54C10,
54D35 | MR 4542795 | Zbl 07655806 | DOI: 10.14712/1213-7243.2022.024
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Keywords:
compactification; coreflection; atom in a lattice
compactification; coreflection; atom in a lattice
Summary:
We define ``the category of compactifications'', which is denoted {\bf{CM}}, and consider its family of coreflections, denoted {\bf{corCM}}. We show that {\bf{corCM}} is a complete lattice with bottom the identity and top an interpretation of the Čech--Stone $\beta$. A $c \in${\bf{corCM}} implies the assignment to each locally compact, noncompact $Y$ a compactification minimum for membership in the ``object-range'' of $c$. We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms in {\bf{corCM}} (thus {\bf{corCM}} is not a set), show that any $c \in${\bf{corCM}} not the identity is above an atom, and that $\beta$ is not the supremum of atoms.
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