Research and Innovation Articles
Published 2011-11-23
Keywords
- lattice,
- pretopologies,
- complete lattices frameworks,
- ultratopolo-gies
How to Cite
Páez Díaz, F. A. (2011). About the lattice of pretopologies on an set X. Revista Integración, Temas De matemáticas, 29(2), 127–142. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/2554
Abstract
We show that (Pretop(X), <=), the lattice of pretopologies on an arbitrary set X, always has a framework; we present a characterization of the co-atoms in Pretop(X) in terms of ultratopologies on X.
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References
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[2] Belmandt Z., Manuel de Prétopologie et ses Applications, Hermes, Paris, 1993.
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[8] De Castro R. & Rubiano G., “Esqueletos de retículos completos”, Bol. Mat. (N. S.), Vol 10, No. 2 (2004), 109–131.
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[10] Kent D.C., “Convergence funtions and their related topologies”, Fund. Math. 54 (1964), 125–133.
[11] Kent D.C., “A note on pretopologies”, Fund. Math. 62 (1968), 95–100.
[12] Mammass D., Djezeri S. and Nouboud F., “A pretopological approach for image segmentation and edge detection”, J. Math. Imaging Vision, Vol. 15, Issue 3 (November 2001),169–179.
[13] Sambin G., “Pretopologies and completeness proofs”, J. Simbolic Logic, Vol 60, No. 3, (September 1995), 861–878.
[14] Stadler B.M.R., Stadler P.F., Wagner G.P. and Fontana W., “The topology of the possible: formal spaces underlying patterns of evolutionary change”, J. Theoret. Biol. 213 (2001), 241–274.
[2] Belmandt Z., Manuel de Prétopologie et ses Applications, Hermes, Paris, 1993.
[3] Carstens A.M., “The laticce of pretopologies on an arbitrary set S”, Pacific J. Math., Vol.29, No. 1 (1969), 67–71.
[4] Cech E., ˇ Topological Spaces, John Wiley & Sons, London, 1966.
[5] Choquet G., “Convergences”, Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.) 23 (1947-48), 57–112.
[6] Davey B.A. and Priestley H.A., Introduction to Laticces and Order, 2nd edition, Cambridge University Press, 2002.
[7] De Castro R. & Rubiano G., “Una revisión del completamiento de Dedekind-MacNeille”, Miscelánea Matemática, Sociedad Matematica Mexicana, 37 (2003), 65–76.
[8] De Castro R. & Rubiano G., “Esqueletos de retículos completos”, Bol. Mat. (N. S.), Vol 10, No. 2 (2004), 109–131.
[9] Fr¨ohlich O., “Das Halbordnungssystem der topologischen Ra¨ume auf einer Menge”, Math. Ann. 156 (1964), 79–95.
[10] Kent D.C., “Convergence funtions and their related topologies”, Fund. Math. 54 (1964), 125–133.
[11] Kent D.C., “A note on pretopologies”, Fund. Math. 62 (1968), 95–100.
[12] Mammass D., Djezeri S. and Nouboud F., “A pretopological approach for image segmentation and edge detection”, J. Math. Imaging Vision, Vol. 15, Issue 3 (November 2001),169–179.
[13] Sambin G., “Pretopologies and completeness proofs”, J. Simbolic Logic, Vol 60, No. 3, (September 1995), 861–878.
[14] Stadler B.M.R., Stadler P.F., Wagner G.P. and Fontana W., “The topology of the possible: formal spaces underlying patterns of evolutionary change”, J. Theoret. Biol. 213 (2001), 241–274.