Compactification of closed preordered spaces

E. Minguzzi

doi:10.4995/agt.2012.1630

Authors

E. Minguzzi Università degli Studi di Firenze

DOI:

https://doi.org/10.4995/agt.2012.1630

Keywords:

Nachbin compactification, Quasi-uniformizable space, completely regularly ordered space

Abstract

A topological preordered space admits a Hausdorff T2-preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff T2-preorder compactification for these spaces and clarify its relation with Nachbin’s compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff T2-preorder compactification is considered.

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Author Biography

E. Minguzzi, Università degli Studi di Firenze

Dipartimento di Matematica Applicata "G. Sansone", Università degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy

References

R. Budic and R. K. Sachs, Causal boundaries for general relativistic spacetimes, J. Math. Phys. 15 (1974), 1302–1309. http://dx.doi.org/10.1063/1.1666812

T. H. Choe and Y. S. Park, Wallman’s type order compactification, Pacific J. Math. 82 (1979), 339–347. http://dx.doi.org/10.2140/pjm.1979.82.339

R. Engelking, General Topology, Berlin: Helderman Verlag (1989).

P. Fletcher and W. Lindgren, Quasi-uniform spaces, vol. 77 of Lect. Notes in Pure and Appl. Math., New York: Marcel Dekker, Inc. (1982).

J. L. Flores, The causal boundary of spacetimes revisited, Commun. Math. Phys. 276 (2007), 611–643. http://dx.doi.org/10.1007/s00220-007-0345-9

R. Geroch, E. H. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. Roy. Soc. Lond. A 237 (1972), 545–567. http://dx.doi.org/10.1098/rspa.1972.0062

S. G. Harris, Universality of the future chronological boundary, J. Math. Phys. 39 (1998), 5427–5445. http://dx.doi.org/10.1063/1.532582

S.W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge: Cambridge University Press (1973). ttp://dx.doi.org/10.1017/CBO9780511524646

D. C. Kent, On the Wallman order compactification, Pacific J. Math. 118 (1985), 159–163. http://dx.doi.org/10.2140/pjm.1985.118.159

D. C. Kent, D. Liu and T. A. Richmond, On the Nachbin compactification of products of totally ordered spaces, Internat. J. Math. & Math. Sci. 18 (1995), 665–676. http://dx.doi.org/10.1155/S0161171295000858

D. C. Kent and T. A. Richmond, Separation properties of the Wallman ordered compactification, Internat. J. Math. & Math. Sci. 13 (1990), 209–222. http://dx.doi.org/10.1155/S0161171290000321

D. C. Kent and T. A. Richmond, A new ordered compactification, Internat. J. Math. & Math. Sci. 16 (1993), 117–124. http://dx.doi.org/10.1155/S0161171293000146

H.-P. A. K¨unzi, Minimal order compactifications and quasi-uniformities, Berlin: Akademie-Verlag, vol. Recent Developments of General Topology and its Applications of Mathematical Research 67 (1992), pages 181–186.

H.-P. A. K¨unzi, A. E. McCluskey and T. A. Richmond, Ordered separation axioms and the Wallman ordered compactification, Publ. Math. Debrecen 73 (2008), 361–377.

D. M. Liu and D. C. Kent, Ordered compactifications and families of maps, Internat. J. Math. & Math. Sci. 20 (1997), 105–110. http://dx.doi.org/10.1155/S016117129700015X

T. McCallion, Compactifications of ordered topological spaces, Proc. Camb. Phil. Soc. 71 (1972), 463–473. http://dx.doi.org/10.1017/S030500410005074X

S. D. McCartan, Separation axioms for topological ordered spaces, Proc. Camb. Phil. Soc. 64 (1968), 965–973. http://dx.doi.org/10.1017/S0305004100043668

E. Minguzzi, The causal ladder and the strength of K-causality. II, Class. Quantum Grav. 25 (2008), 015010. http://dx.doi.org/10.1088/0264-9381/25/1/015010

E. Minguzzi, Normally preordered spaces and utilities, Order, to appear. http://dx.doi.org/10.1007/s11083-011-9230-4

E. Minguzzi, Quasi-pseudo-metrization of topological preordered spaces, Topol. Appl. 159 (2012), 2888–2898. http://dx.doi.org/10.1016/j.topol.2012.05.029

E. Minguzzi, Topological conditions for the representation of preorders by continuous utilities, Appl. Gen. Topol. 13 (2012), 81–89.

L. Nachbin, Topology and order, Princeton: D. Van Nostrand Company, Inc. (1965).

S. Nada, Studies on Topological Ordered Spaces, Ph.D. thesis, Southampton (1986).

R. Penrose, Conformal treatment of infinity, New York: Gordon and Breach, vol. Relativity, Groups and Topology, pages 563–584 (1964).

I. Rácz, Causal boundary of space-times, Phys. Rev. D 36 (1987), 1673–1675. http://dx.doi.org/10.1103/PhysRevD.36.1673

I. Rácz, Causal boundary for stably causal spacetimes, Gen. Relativ. Gravit. 20 (1988), 893–904. http://dx.doi.org/10.1007/BF00760089

T. A. Richmond, Posets of ordered compactifications, Bull. Austral. Math. Soc. 47 (1993), 59–72. http://dx.doi.org/10.1017/S0004972700012260

S. Scott and P. Szekeres, The abstract boundary: a new approach to singularities of manifolds, J. Geom. Phys. 13 (1994), 223–253. http://dx.doi.org/10.1016/0393-0440(94)90032-9

H. Seifert, The causal boundary of space-times, Gen. Relativ. Gravit. 1 (1971), 247–259. http://dx.doi.org/10.1007/BF00759536

L. B. Szabados, Causal boundary for strongly causal spacetimes, Class. Quantum Grav. 5 (1988), 121–134. http://dx.doi.org/10.1088/0264-9381/5/1/017

L. B. Szabados, Causal boundary for strongly causal spacetimes II, Class. Quantum Grav. 6 (1989), 77–91. http://dx.doi.org/10.1088/0264-9381/6/1/007

S. Willard, General topology, Reading: Addison-Wesley Publishing Company (1970).