Sobriety of crisp and fuzzy topological spaces

The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces

Sobriety of crisp and fuzzy topological spaces

The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces

Lebesgue quasi-uniformity on textures

[EN] This is a continuation of the work where the notions of Lebesgue uniformity and Lebesgue quasi uniformity in a texture space were introduced. It is  well known that the quasi uniform space with a compact topology has the Lebesgue property. This result is extended to direlational quasi uniformities and dual dicovering quasi uniformities. Additionally we discuss the completeness of lebesgue di-uniformities and dual dicovering lebesgue di-uniformities.Ozcag, S. (2015). Lebesgue quasi-uniformity on textures. Applied General Topology. 16(2):167-181. doi:10.4995/agt.2015.3323.SWORD167181162Brown, L. M., Ertürk, R., & Dost, Ş. (2004). Ditopological texture spaces and fuzzy topology, I. Basic concepts. Fuzzy Sets and Systems, 147(2), 171-199. doi:10.1016/j.fss.2004.02.009L. M. Brown and M. M. Gohar, Compactness in Ditopological Texture Spaces}, Hacettepe journal of Mathematics and Statistics 38, no. 1 (2009), 21--43.P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Marcel Dekker, (New York and Basel, 1982).Gantner, T. E., & Steinlage, R. C. (1972). Characterizations of Quasi-Uniformities†. Journal of the London Mathematical Society, s2-5(1), 48-52. doi:10.1112/jlms/s2-5.1.48Hutton, B. (1977). Uniformities on fuzzy topological spaces. Journal of Mathematical Analysis and Applications, 58(3), 559-571. doi:10.1016/0022-247x(77)90192-5J. Marin and S. Romaguera, On quasi uniformly continuous functions and Lebesgue spaces}, Publicationes Mathematicae Debrecen 48 (1996), 347-355.S. Özcag, F. Yildiz and L. M. Brown, Convergence of regular difilters and the completeness of di-uniformities, Hacettepe Journal of mathematics and statistics 34, (2005) 53-68

On φ 1,2-countable compactness and filters

[EN] In this work the author investigates some relations between φ 1,2-countable compactness, filters, sequences and φ 1,2-closure operators.Yalvaç, T. (2003). On φ 1,2-countable compactness and filters. Applied General Topology. 4(1):35-46. doi:10.4995/agt.2003.2008.SWORD354641A. S. Mashhour, A. A. Allam, F. S. Mahmoud and F. H. Khedr, On supratopological spaces, Indian J. Pure Appl. Math. 14 (4) (1983), 502-510.M. E. Abd El-Monsef and E. F. Lashien, Local discrete extensions of supratopologies, Tamkong J. Math 31 (1) (1990), 1-6.R. F. Dickmann Jr. and J. R. Porter, θ-perfect and absolutely closed functions, Illinois J. Math. 21 (1977), 42-60.Dickman, R. F., & Krystock, R. L. (1980). S-Sets and S-Perfect Mappings. Proceedings of the American Mathematical Society, 80(4), 687. doi:10.2307/2043452G. Di Maio and T. Noiri, On s-closed spaces, Indian J. Pure Appl. Math. 18 (3) (1987), 226-233.K. Dlaska, N. Ergun and M. Ganster, Countably S-closed spaces, Math. Slovaca 44 (3) (1994), 337-348.J. Dontchev and M. Ganster, On covering spaces with semi-regular sets, Ricerche Math. 45 (1996), 229-245.J. Dugundji, Topology, (Allyn and Bacon, Boston, Mass., 1966).Herrmann, R. A. (1979). rc-convergence. Proceedings of the American Mathematical Society, 75(2), 311-311. doi:10.1090/s0002-9939-1979-0532157-5Herrington, L. L., & Long, P. E. (1975). Characterizations of H-Closed Spaces. Proceedings of the American Mathematical Society, 48(2), 469. doi:10.2307/2040285Herrington, L. L. (1976). Remarks on H(i) Spaces and Strongly-Closed Graphs. Proceedings of the American Mathematical Society, 58(1), 277. doi:10.2307/2041400A. Kandil, E. E. Kerre and A. A. Nouh,Operations and mappings on fuzzy topological spaces, Ann. Soc. Sci. Bruxelles 105 (4) (1991), 165-168.S. N. Maheshwari and S. S. Thakur, Jour. Sci. Res. 3 (1981), 121-123.S. N. Maheshwari and S. S. Thakur, On α-compact spaces, Bull. Inst. Math. Academia Sinica 13 (4) (1985), 341-347.T. Noiri, On RS-compact spaces, J. Korean Math. Soc. 22 (1) (1985), 19-34.T. G. Raghavan, On H(1)-closed spaces-II, Bull. Cal. Math. Soc. 77 (1985), 171-180.Stephenson, R. M. (1968). Pseudocompact spaces. Transactions of the American Mathematical Society, 134(3), 437-437. doi:10.1090/s0002-9947-1968-0232349-6T. Thompson, SQ-closed spaces, Math. Japonica 22 (4) (1977), 491-495.D. Thanapalan and T. G. Raghavan, On strongly H(1)-closed spaces, Bull. Cal. Math. Soc. 76 (1984), 370-383.T. H. Yalvaç, A unified approach to compactness and filters, Hacettepe Bull. Nat. Sci. Eng., Series B 29 (2000), 63-75.T. H. Yalvaç, On some unifications (Presented at The First Turkish International Conference on Topology and Its Applications, Istanbul, 2000), Hacettepe Bull. Nat. Sci. Eng., Series B 30 (2001), 27-38.T. H. Yalvaç, A unified theory on some basic topological concepts, International Conference on Topology and its Applications, Macedonia, (2000). T. H. Yalvaç, Unifications of some concepts related to the Lindelöf property, submitted

Generalisations of filters and uniform spaces

The notion of a filter F ∈ 2²x has been extended to that of a : prefilter: ƒ ∈ 1²x, generalised filter ƒ ∈ 2²x x and fuzzy filter ᵩ ∈ 1¹x. A uniformity is a filter with some other conditions and the notion of a uniformity D ∈ 2²xxx has been extended to that of a : fuzzy uniformity d ∈ 1²xxx , generalised uniformity ∈ 1²xxx and super uniformity b ∈ 1¹x. We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces