Abstract
In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).
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References
B. Banaschewski: Injectivity and essential extensions in equational classes of algebras,Proc. Conf. Univ. Algebra 1969, Queen's Univ., Kingston 1970.
B. Banaschewski: Projective covers in categories of topological spaces and topological algebras,Proc. Kanpur Topol. Conf. 1968, Academia Prague 1071, 63–91.
B. Banaschewski: Essential extensions ofT 0-spaces,Gen. Topology Appl. 7 (1977), 233–246.
R.-E. Hoffmann: Essential extensions ofT 1-spaces,Canad. Math. Bull. 24 (1981), 237–240.
R.-E. Hoffmann:Continuous Posets, Prime Spectra of Completely Distribute Complete Lattices, and Hausdorff Compactifications, Springer Lecture Notes Math.871 (1981), 159–208.
F. Obreanu: Espaces localement absolument fermés,Ann. Acad. Repub. Pop. Romane, Sect. Sti. Fiz. Chim., Ser. A3 (1950), 375–394.
J. Porter: On locally H-closed spaces,Proc. London Math. Soc. (3 20 (1970), 193–204.
J. Porter: Lattices of H-closed extensions,Bull L'Acad. Polon. Sci., Ser. sci. math., astr. et phys. 22 (1974), 831–837.
D. Scott:Continuous Lattices, Springer Lecture Notes Math274 (1972), 97–136.
O. Wyler: Injective spaces and essential extensions in TOP,Gen. Topology Appl. 7 (1977), 247–249.
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Dedicated to my friend Nico Pumplün on his sixtieth birthday
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Herrlich, H. Essential extensions of Hausdorff spaces. Appl Categor Struct 2, 101–105 (1994). https://doi.org/10.1007/BF00878506
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DOI: https://doi.org/10.1007/BF00878506