Previous |  Up |  Next


Full entry | Fulltext not available (moving wall 24 months)      Feedback
compactification; coreflection; atom in a lattice
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.
[1] Birkhoff G.: Lattice Theory. American Mathematical Society Colloquium Publications, 25, American Mathematical Society, Providence, 1979. MR 0598630 | Zbl 0537.06001
[2] Carrera R., Hager A. W.: A classification of hull operators in archimedean lattice-ordered groups with unit. Categ. Gen. Algebr. Struct. Appl. 13 (2020), no. 1, 83–103. MR 4162033
[3] Chandler R. E.: Hausdorff Compactifications. Lecture Notes in Pure and Applied Mathematics, 23, Marcel Dekker, New York, 1976. MR 0515002
[4] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[5] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathematics, 43, Springer, New York, 1976. MR 0407579 | Zbl 0327.46040
[6] Hager A. W.: Minimal covers of topological spaces. Ann. New York Acad. Sci., 552, New York Acad. Sci., New York, 1989, pages 44–59. MR 1020773
[7] Hager A. W., Martinez J.: Hulls for various kinds of $\alpha$-completeness in Archimedean lattice-ordered groups. Order 16 (1999), no. 1, 89–103. DOI 10.1023/A:1006323031986 | MR 1740743
[8] Hager A. W., Wynne B.: Atoms in the lattice of covering operators in compact Hausdorff spaces. Topology Appl. 289 (2021), Paper No. 107402, 9 pages. DOI 10.1016/j.topol.2020.107402 | MR 4192355
[9] Herrlich H.: Topologische Reflexionen und Coreflexionen. Lecture Notes in Mathematics, 78, Springer, Berlin, 1968 (German). MR 0256332 | Zbl 0182.25302
[10] Herrlich H., Strecker G. E.: Category Theory. Sigma Series in Pure Mathematics, 1, Heldermann Verlag, Berlin, 1979. MR 0571016 | Zbl 1125.18300
[11] Magill K. D., Jr.: $N$-point compactifications. Amer. Math. Monthly 72 (1965), 1075–1081. DOI 10.1080/00029890.1965.11970675 | MR 0185572
[12] Porter J. R., Woods R. G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York, 1988. MR 0918341 | Zbl 0652.54016
Partner of
EuDML logo