Lattices of topological extensions

Mack, John; Rayburn, Marlon; Woods, Grant

doi:10.1090/S0002-9947-1974-0350700-6

Lattices of topological extensions
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by John Mack, Marlon Rayburn and Grant Woods PDF

Trans. Amer. Math. Soc. 189 (1974), 163-174 Request permission

Abstract:

For completely regular Hausdorff spaces, we consider topological properties P which are akin to compactness in the sense of Herrlich and van der Slot and satisfy the equivalent of Mrowka’s condition (W). The algebraic structure of the family of tight extensions of X (which have P and contain no proper extension with that property) is studied. Where X has P locally but not globally, the relations between the complete lattice ${P^ \ast }(X)$ of those tight extensions which are above the maximal one-point extension and the topology of the P-reflection are investigated and conditions found under which ${P^\ast }(X)$ characterizes $\gamma X - X$. The results include those of Magill on the lattice of compactifications of a locally compact space, and other applications are considered.

References

N. Boboc and Gh. Sireţchi, Sur la compactification d’un espace topologique, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.) 5(53) (1961), 155–165 (1964) (French). MR 182948
W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107–118. MR 222846, DOI 10.1090/S0002-9947-1968-0222846-1
Nancy Dykes, Mappings and realcompact spaces, Pacific J. Math. 31 (1969), 347–358. MR 263039, DOI 10.2140/pjm.1969.31.347
Zdeněk Frolík, A generalization of realcompact spaces, Czechoslovak Math. J. 13(88) (1963), 127–138 (English, with Russian summary). MR 155289, DOI 10.21136/CMJ.1963.100555
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
S. L. Gulden, W. M. Fleischman, and J. H. Weston, Linearly ordered topological spaces, Proc. Amer. Math. Soc. 24 (1970), 197–203. MR 250272, DOI 10.1090/S0002-9939-1970-0250272-2
H. Herrlich and J. van der Slot, Properties which are closely related to compactness, Nederl. Akad. Wetensch. Proc. Ser. A 70=Indag. Math. 29 (1967), 524–529. MR 0222848, DOI 10.1016/S1385-7258(67)50068-9
John Mack, Marlon Rayburn, and Grant Woods, Local topological properties and one point extensions, Canadian J. Math. 24 (1972), 338–348. MR 295297, DOI 10.4153/CJM-1972-028-1
Kenneth D. Magill Jr., The lattice of compactifications of a locally compact space, Proc. London Math. Soc. (3) 18 (1968), 231–244. MR 229209, DOI 10.1112/plms/s3-18.2.231
S. Mrówka, Some comments on the author’s example of a non-${\cal R}$-compact space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 443–448 (English, with Russian summary). MR 268852
Z. Frolík and S. Mrówka, Perfect images of ${\cal R}$- and ${\cal N}$-compact spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 369–371 (English, with Russian summary). MR 307167
S. Mrówka, On local topological properties, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 951–956 LXXX (English, with Russian summary). MR 0096184
Nancy M. Warren, Properties of Stone-Čech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599–606. MR 292035, DOI 10.1090/S0002-9939-1972-0292035-X
R. Grant Woods, Some $\aleph _{O}$-bounded subsets of Stone-Čech compactifications, Israel J. Math. 9 (1971), 250–256. MR 278266, DOI 10.1007/BF02771590