TOWARDS THE THEORY OF L-BORNOLOGICAL SPACES

Document Type : Research Paper

Authors

1 Institute of Pure Mathematics, University of Tartu, J.Liivi street 2, EE-50409 Tartu, Estonia

2 Department of Mathematics, University of Latvia, Zellu street 8, LV-1002, Riga, Latvia and Institute of Mathematics and CS, University of Latvia, Raina bulv. 29, LV-1586, Riga, Latvia

Abstract

The concept of an $L$-bornology is introduced and the theory of $L$-bornological spaces
is being developed. In particular the lattice of all $L$-bornologies on a given set is studied and basic properties of
the category of $L$-bornological spaces and bounded mappings are investigated.

Keywords


bibitem{AHS}J. Ad'amek, H. Herrlich and G. E. Strecker, {it  Abstract and
concrete categories}, John Wiley & Sons, New York, 1990.
bibitem{Beer} G. Beer, {it Metric bornologies and Kuratowska-Painleve convergence to the empty set},
{Journal of Convex Analysis}, {bf 8} (2001), 273-289.
%bibitem{Beer_Levi} Gerald Beer, Levi
bibitem{Bir} G. Birkhoff, {it Lattice theory}, AMS Providence, RI,
1995.
bibitem{Ch68} C. L. Chang, {it Fuzzy topological spaces}, {J. Math. Anal.
Appl.}, {bf 24} (1968), 182-190.
bibitem{Gierz} G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, {it  Continuous lattices and domains},
Cambridge University Press,  Cambridge, 2003.
bibitem{Go67} J. A. Goguen, {it $L$-fuzzy sets}, {J. Math. Anal. Appl.}, {bf 18} (1967) 145-174.
bibitem{Go73} J. A. Goguen, {it The fuzzy Tychonoff theorem}, { J. Math. Anal. Appl.}, {bf 18} (1973), 734-742.
bibitem{HS} H. Herrlich and G. E. Strecker, { it Category theory}, Heldermann Verlag, Berlin, 1987.

bibitem{HN} H. Hogbe-Nlend, {it Bornology and functional analysis}, Math. Studies, North-Holland, Amsterdam, {bf 26} (1977).
bibitem{HoSo99} U. H"ohle and A. v{S}ostak, {it Axiomatics for fixed-based fuzzy topologies}, Chapter 3 in [8].
bibitem{HuS1} S. T. Hu, {it Boundedness in a topological space}, {it J. Math. Pures Appl.}, {bf 28} (1949), 287-320.
bibitem{HuS2} S. T. Hu, {it Introduction to general topology}, Holden-Day, San-Francisko, 1966.
bibitem{Hut} B. Hutton, {it Normality in fuzzy topologicl spaces}, {J. Math. Anal. Appl.}, {bf 50} (1975), 74-79.
bibitem{JiangYan} S. Q. Jiang and C. H. Yan, { it Fuzzy bounded sets and totally fuzzy bounded sets in I-topological vector spaces},
{ Iranian Journal of Fuzzy Systems}, {bf 6(3)} (2009), 73-90.
bibitem{MFS} U. H"ohle and S. E. Rodabaugh, eds., {it Mathematics of fuzzy sets: logic, topology and
measure theory}, Handbook Series, Kluwer Acad. Publ., {bf3} (1999).
bibitem{NVT} A. Narayaanan, S. Vijayabalaji and N. Thillaigovitidan, {it Intuitionistic fuzzy linear bounded operators},
{Iranian Journal of Fuzzy Systems}, {bf 4(1)} (2007), 89-93.
bibitem{NR} C. V. Negoita and D. A. Ralescu, { it Application of fuzzy sets to system analysis}, John Wiley & Sons,
New York, 1975.
bibitem{Ro99} S. E. Rodabaugh, {it Powerset operator foundations for poslat fuzzy set theories and topologies},
Chapter 2 in
cite{MFS}.
bibitem{Za} L. Zadeh, {it Fuzzy sets}, {Information and Control}, {bf 8} (1965), 338-353.