F-door spaces and F-submaximal spaces

Authors

  • Lobna Dridi University of Tunis
  • Sami Lazaar University Tunis-El Manar
  • Tarek Turki University Tunis-El Manar

DOI:

https://doi.org/10.4995/agt.2013.1621

Keywords:

Categories, Functors, Door spaces, Submaximal spaces, Primal spaces

Abstract

Submaximal spaces and door spaces play an enigmatic role in topology. In this paper, reinforcing this role, we are concerned with reaching two main goals:

The first one is to characterize topological spaces X such that F(X) is a submaximal space (resp., door space) for some covariant functor Ff rom the category Top to itself. T0, and FH functors are completely studied.

Secondly, our interest is directed towards the characterization of maps f given by a flow (X, f) in the category Set, such that (X,P(f)) is submaximal (resp., door) where P(f) is a topology on X whose closed sets are exactly the f-invariant sets.

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Author Biographies

Lobna Dridi, University of Tunis

Department of Mathematics, Tunis Preparatory Engineering Institute. University of Tunis. 1089 Tunis, Tunisia.

Sami Lazaar, University Tunis-El Manar

Department of Mathematics, Faculty of Sciences of Tunis. University Tunis-El Manar. “Campus Universitaire” 2092 Tunis, Tunisia.

Tarek Turki, University Tunis-El Manar

Department of Mathematics, Faculty of Sciences of Tunis. University Tunis-El Manar. “Campus Universitaire” 2092 Tunis, Tunisia.

References

C. E. Aull and W. J. Thron, Separation axioms between T0 and T1, Indag. Math. 24 (1962), 26–37.

K. Belaid, L. Dridi and O. Echi, Submaximal and door compactifications, Topology Appl. 158 (2011), 1969–1975. http://dx.doi.org/10.1016/j.topol.2011.06.039

K. Belaid, O. Echi and R. Gargouri, A-spectral spaces, Topology Appl. 138 (2004), 315 − 322. http://dx.doi.org/10.1016/j.topol.2003.08.009

K. Belaid, O. Echi and S. Lazaar, T(_,_)-spaces and the Wallman compactification, Int. J. Math. Math. Sc. 68 (2004), 3717–3735. http://dx.doi.org/10.1155/S0161171204404050

G. Bezhanishvili, L. Esakia and D. Gabelaia, Some results on modal axiomatization and definability for topological spaces, Studia Logica. 81 (2005), 325–355. http://dx.doi.org/10.1007/s11225-005-4648-6

E. Bouacida, O. Echi, G. Picavet and E. Salhi, An extension theorem for sober spaces and the Goldman topology, Int. J. Math. Math. Sc. 2003, no. 51 (2003), 3217–3239. http://dx.doi.org/10.1155/S0161171203212230

N. Bourbaki, Eléments de mathématiques, topologie générale, chapitres 1 à 4, 1990.

N. Bourbaki, Topologie générale, chapitres , 3rd ed., Actualités Scientifiqueqs et industrielles 1142 (Hermann, Paris, 1961).

C. Cassidy, M. Hebert and J. M. Kelly, Reflective subcategories, localization and factorization systems, J. Austral. Math. Soc (Ser. A) 38 (1985), 387–429. http://dx.doi.org/10.1017/S1446788700023624

Y.S. Cho, On reflective subcategories, Kyungpook Math. J. 18 (1978), 143–146.

J. Cincura, Closed structures on reflective subcategories of the category of topological spaces, Topology Appl. 37 (1990), 237–247. http://dx.doi.org/10.1016/0166-8641(90)90022-T

A. S. Davis, Indexed systems of neighborhoods for general topological spaces, Am. Math. Mon. 68 (1961), 886–894. http://dx.doi.org/10.2307/2311686

C. Dorsett, Characterizations of spaces using T0-identification spaces, Kyungpook Math. J. 17 (1977), 175–179.

D. Drake and W. J. Thron, On the representations of an abstract lattice as the family of closed sets on a topological space, Trans. Amer. Math. Soc. 120, no. 2 (1965), 57–71. http://dx.doi.org/10.1090/S0002-9947-1965-0188963-7

O. Echi, Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6, no. 2 (2003), 489–507.

O. Echi, The category of flows of Set and Top, Topology Appl. 159 (2012), 2357–2366. http://dx.doi.org/10.1016/j.topol.2011.11.059

O. Echi, R. Gargouri and S. Lazaar, On the Hochster dual of topological space, Topology Proc. 32 (2008), 153–166.

O. Echi and S. Lazaar, Reflective subcategories, Tychonoff spaces, and spectral spaces, Topology Proc. 34 (2009), 307–319.

O. Echi and S. Lazaar, Universal spaces, Tychonoff and spectral spaces, Math. Proc. R. Ir. Acad. 109 (2009), 35–48. http://dx.doi.org/10.3318/PRIA.2008.109.1.35

P. D. Finch, On the lattice-equivalence of topological spaces, J. Austral. Math. Soc. 6 (1966), 495–511. http://dx.doi.org/10.1017/S1446788700004973

A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique I: le langage des schémas, Inst. Hautes Etudes Sci. Publ. Math. no. 4, 1960.

A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, Die Grundlehren der mathematischen Wissenschaften, vol. 166, Springer-Verlag, New York, 1971.

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. http://dx.doi.org/10.1090/S0002-9947-1969-0251026-X

J. L. Kelly, General topology, D. Van Nostrand Company, Inc., Princeton, NJ, 1955.

J. F. Kennisson, The cyclic spectrum of a boolean flow, Theory Appl. Categ. 10 (2002), 392–409.

J. F. Kennisson, Spectra of finitely generated boolean flows, Theory Appl. Categ. 16 (2006), 434–459.

H-P. A. K¨unzi, A. E. Mccluskey and T. A. Richmond, Ordered separation axioms and the Wallman ordered compactification, Pupl. Math. Debrecen 73, no. 3-4 (2008), 361–377.

H-P. A. K¨unzi and T. A. Richmond, Ti-ordered reflections, Appl. Gen. Topol. 6, no. 2 (2005), 207–216.

M. W. Mislove, Topology, domain theory and theoretical computer science, Topology Appl. 89 (1998), 3–59. http://dx.doi.org/10.1016/S0166-8641(97)00222-8

N. A. Shanin, On separation in topological spaces, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 38 (1943), 110–113.

M. H. Stone, Applications of boolean algebra to topology, Mat. Sb. 1 (1936), 765–772.

W. J. Thron, Lattice-equivalence of topological spaces, Duke Math. J. 29 (1962), 671–679. tp://dx.doi.org/10.1215/S0012-7094-62-02968-X

J. W. Tukey, Convergence and uniformity in topology, Annals of Mathematics Studies, no. 2. Princeton University Press, 1940 Princeton, N. J.

P. Uryshon, Uber the m¨achtigkeit der zusammenh¨agenden Mengen, Math. Ann. 94 (1925), 262–295. http://dx.doi.org/10.1007/BF01208659

W. T. Van Est and H. Freudenthal, Trennung durch stetige funktionen in topologischen R¨aumen, Indag. Math. 13 (1951), 359–368.

R. C. Walker, The Stone-Cech compactification, Ergebnisse der Mathamatik Band 83.

Y-M. Wong, lattice-invariant properties of topological spaces, Proc. Amer. Math. Soc. 26, no. 1 (1970), 206–208. http://dx.doi.org/10.1090/S0002-9939-1970-0261549-9

K. W. Yip, Quasi-homeomorphisms and lattice-equivalences of topological spaces, J. Austral. Math. Soc. 14 (1972), 41–44. http://dx.doi.org/10.1017/S1446788700009617

J. W. T. Youngs, A note on separation axioms and their application in the theory of a locally connected topological space, Bull. Amer. Math. Soc. 49 (1943), 383–385. http://dx.doi.org/10.1090/S0002-9904-1943-07922-0

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How to Cite

[1]
L. Dridi, S. Lazaar, and T. Turki, “F-door spaces and F-submaximal spaces”, Appl. Gen. Topol., vol. 14, no. 1, pp. 97–113, Apr. 2013.

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Regular Articles