A model of quotient spaces

Hawete Hattab 1
  • 1 Laboratoire des Systèmes Dynamiques et Combinatoires Université de Sfax Tunisia and Umm Alqura University, Department of Mathematics KSA, , Kairouan, Tunisie

Abstract

Let R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃ℜ̅ by x̃ ̃ℜ̅y if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃ ̃ℜ̅ is called the trajectory class space. In this paper, we show that the space E/̃ ̃ℜ̅ is a simple model of the quotient space E/R. This model can provide a finite model. Some applications to orbit spaces of groups of homeomorphisms and leaf spaces are given.

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  • [1] P.S. Alexandroff. Diskrete Räume. MathematiceskiiSbornik (N.S.) 2 (1937), 501-518.

  • [2] J.A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics 2032, Springer-Verlag Berlin Heidelberg 2011.

  • [3] J.A. Barmak and E.G. Minian. Minimal finite models. J. Homotopy Relat. Struct. 2 No. 1, (2007), 127-140.

  • [4] E.G. Begle, The Vietoris mapping theorem for bicompact spaces. II, Michigan Math. J. 3 (1955/1956) 179-180.

  • [5] G.E. Bredon, Sheaf Theory, second ed., Springer-Verlag, New York, 1997.

  • [6] C. Bonatti, H. Hattab and E. Salhi, Quasi-orbits spaces associated to T0-spaces, Fund. Math. 211 (2011),267-291.

  • [7] C. Bonatti, H. Hattab, E. Salhi and G. Vago, Hasse diagrams and orbit class spaces, Topology and its Applications 158, Issue 6, 1 (2011), 729-740.

  • [8] R. Bott. Lectures on characteristic classes and foliations, Lecture Notes in Mathematics 279, Springer, 1972.

  • [9] E. Bouacida, O. Echi, E. Salhi, Feuilletage et topologie spectrale, J. Math. Soc. Japan Vol.52, No.2 (2000), 447-464.

  • [10] E. Bouacida, O. Echi, E. Salhi, Foliations, Spectral Topology and Special Morphism, Comm. Ring. Theory III Lect.not.Pure.Appl.Math.M.Dekker Vo1.205 (1999), 111-132.

  • [11] J. S. Calcut, R. E. Gompf and J. D. McCarthyc, On fundamental groups of quotient spaces, Topology and its Applications 159 (2012) 322-330.

  • [12] J.A.E. Dieudonné, A History of Algebraic and Differential Topology. 1900-1960, Birkhäuser, Boston, 1989.

  • [13] J. Dydak, G. Kozlowski, Vietoris-Begle theorem and spectra, Proc. Amer. Math. Soc. 113 (1991) 587-592.

  • [14] J. Dydak, J.J. Walsh, Cohomological local connectedness of decomposition spaces, Proc. Amer. Math. Soc. 107 (1989) 1095-1105.

  • [15] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Die Grundlehren der mathematischen Wissenschaften, vol. 166, Springer-Verlag, New York, 1971.

  • [16] C. Godbillon, Feuilletages. Etudes géométriques, Birkauser -Verlag , 1991.

  • [17] A. Haefliger. Feuilletages sur les variétes ouvertes, Topology 9 (1970), 183-194.

  • [18] H. Hattab, Characterization of quasi-orbit spaces, Qualitative theory of dynamical systems,1-8 , January 11, 2012.

  • [19] H. Hattab, E. Salhi, Groups of homeomorphisms and spectral topology, Topology Proceedings Vol. 28, No.2 (2004), 503-526.

  • [20] N. Le Anh, The Vietoris-Begle theorem, Mat. Zametki 35 (1984) 847-854; English translation in Math. Notes 35 (1984) 444-447.

  • [21] J. Leray, Structure de l’anneau d’homologie d’une représentation, C. R. Acad. Sci. Paris 222 (1946) 1419-1422.

  • [22] M.C. McCord. Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33 (1966), 465-474.

  • [23] H. Reitberger, Leopold Vietoris (1891-2002), Notices Amer. Math. Soc. 49 (2002) 1232-1236.

  • [24] E. Salhi. Problème de structure dans les feuilletages de codimension un de classe C0. Thèse d’Etat, publication de l’IRMA Strasbourg, 1984.

  • [25] R.E. Stong, Finite topological spaces. Trans. Amer. Math. Soc. 123 (1966), 325-340.

  • [26] S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957) 604-610.

  • [27] D.F. Snyder, A characterization of sheaf-trivial, proper maps with cohomologically locally connected images, Topology Appl. 60 (1994) 75-85.

  • [28] L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927) 454-472.

  • [29] E. Wofsey, On the algebraic topology of finite spaces (2008) http://www.math.harvard.edu/~waffle/finitespaces.pdf.

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