Abstract
In this paper, we study the functorial behaviour of the Wallman–Shanin-type compactification for weakly symmetric T 1 approach spaces, as defined in (R. Lowen and M. Sioen [13]). We do this by generalizing the technique used by H. L. Bentley and S. A. Naimpally in their paper [2], yielding a recharacterization of our quantified compactification theory in terms of contiguity clusters of [0, ∞]-valued functionals and we also define a construct WS (which is the counterpart of SEP in [2]) on a suitable non-full subconstruct of which the Wallman–Shanin-type compactification determines an epireflection.
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References
Adámek, J., Herrlich, H., and Strecker, G.: Abstract and Concrete Categories, Wiley, New York, 1990.
Bentley, H. L. and Naimpally, S. A.: Wallman T 1 compactifications as epireflections, Gen.Topol. Appl. 4 (1974), 29–41.
Frink, O.: Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602–607.
Gagrat, M. S. and Naimpally, S. A.: Wallman compactifications and Wallman realcompactifications, J. Austral. Math. Soc. 15 (1973), 417–427.
Harris, D.: The Wallman compactification as a functor, Gen. Topol. Appl. 1 (1971), 273–281.
Harris, D.: TheWallman compactification is an epireflection, Proc. Amer. Math. Soc. 31 (1972) 265–267.
Herrlich, H.: Categorical topology 1971–1981, Universität Bremen, Mathematik Arbeitspapiere 24 A, Math. Forschungspapiere, 1981, pp. 1–105.
Herrlich, H.: —Topologische reflexionen und coreflexionen, Lecture Notes in Mathematics 78, Springer-Verlag, Berlin, 1968.
Hušek, M.: Čech-Stone-like compactifications for general topological spaces, Comment. Math. Univ. Carolin. 33(1) (1992), 159–163.
Lowen, R.: Approach spaces: A common supercategory of TOP and MET, Math. Nachr. 141 (1989), 183–226.
Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997.
Lowen, R. and Robeys, K.: Compactifications of products of metric spaces and their relations to Čech-Stone and Smirnov compactifications, Topol. Appl. 55 (1994), 163–183.
Lowen, R. and Sioen, M.: A Wallman-Shanin-type compactification for approach spaces, submitted.
Nagata, J.: Modern General Topology, North-Holland Mathematical Library, Vol. 33, North-Holland, Amsterdam, 1985.
Naimpally, S. A. and Warrack, B. D.: Proximity Spaces, Camebridge Tracts in Math. and Math. Phys., Vol. 59, Cambridge University Press, 1970.
Preuβ, G.: Theory of Topological Structures: An Approach to Categorical Topology, Mathematics and its Applications, Reidel, Dortrecht, 1988.
Sioen, M.: The Čech-Stone compactification for Hausdorff uniform approach spaces is of Wallman-Shanin-type, to appear in Questions Answers Gen. Topology 16(2).
Steiner, E. F.: Wallman spaces and compactifications, Fund. Math. LXI (1968), 295–304
Walker, R. C.: The Stone-Čech Compactification, Springer-Verlag, Berlin, 1974.
Wallman, H.: Lattices and topological spaces, Ann. Math. 39 (1938), 112–126.
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Sioen, M. On the Epireflectivity of the Wallman–Shanin-Type Compactification for Approach Spaces. Applied Categorical Structures 8, 607–637 (2000). https://doi.org/10.1023/A:1008670621730
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DOI: https://doi.org/10.1023/A:1008670621730