Some Comments on Q-Irresolute and Quasi-Irresolute Functions

The aim of this paper is to continue the study of θ-irresolute and quasi-irresolute functions as well as to give an example of a function which is θ-irresolute but neither quasi-irresolute nor an R-map and thus give an answer to a question posed by Ganster, Noiri and Reilly. We prove that RS-compactness is preserved under open, quasi-irresolute surjections

On some applications of fuzzy points

[EN] The notion of preopen sets play a very important role in General Topology and Fuzzy Topology. Preopen sets are also called nearly open and locally dense. The purpose of this paper is to give some applications of fuzzy points in fuzzy topological spaces. Moreover, in section 2 we offer some properties of fuzzy preclosed sets through the contribution of fuzzy points and we introduce new separation axioms in fuzzy topological spaces. Also using the notions of weak and strong fuzzy points, we investigate some properties related to the preclosure of such points, and also their impact on separation axioms. In section 3, using the notion of fuzzy points, we introduce and study the notions of fuzzy pre-upper limit, fuzzy pre-lower limit and fuzzy pre-limit. Finally in section 4, we introduce the fuzzy pre-continuous convergence on the set of fuzzy pre-continuous functions and give a characterization of the fuzzy pre-continuous convergence through the assistance of fuzzy pre-upper limit.S. P. Moshokoa has been supported by the South African National Research Foundation under grant number 2053847. Also, the authors thank the referee for making several suggestions which improved the quality of this paper.Ganster, M.; Georgiou, D.; Jafari, S.; Moshokoa, S. (2015). On some applications of fuzzy points. Applied General Topology. 6(2):119-133. https://doi.org/10.4995/agt.2005.1951SWORD1191336

δ-closure, θ-closure and generalized closed sets

[EN] We study some new classes of generalized closed sets (in the sense of N. Levine) in a topological space via the associated δ-closure and θ-closure. The relationships among these new classes and existing classes of generalized closed sets are investigated. In the last section we provide an extensive and more or less complete survey on separation axioms characterized via singletons.Cao, J.; Ganster, M.; Reilly, IL.; Steiner, M. (2005). δ-closure, θ-closure and generalized closed sets. Applied General Topology. 6(1):79-86. doi:10.4995/agt.2005.1964.798661Cao, J., Ganster, M., & Reilly, I. (2002). On generalized closed sets. Topology and its Applications, 123(1), 37-46. doi:10.1016/s0166-8641(01)00167-5J. Cao, M. Ganster and I. Reilly, Submaximality, extremal disconnectedness and generalized closed sets, Houston J. Math. 24 (1998), 681-688.Cao, J., Greenwood, S., & Reilly, I. L. (2001). Generalized closed sets: a unified approach. Applied General Topology, 2(2), 179. doi:10.4995/agt.2001.2148K. Dlaska and M. Ganster, S-sets and co-S-closed topologies, Indian J. Pure Appl. Math. 23 (1992), 731-737.J. Dontchev and M. Ganster, On δ-generalized closed sets and T3/4 spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math. 17 (1996), 15-31.Dontchev, J., & Maki, H. (1999). Onθ-generalized closed sets. International Journal of Mathematics and Mathematical Sciences, 22(2), 239-249. doi:10.1155/s0161171299222399W. Dunham, T1/2-spaces, Kyungpook Math. J. 17 (1977), 161-169.D. Jankovic, On some separation axioms and θ-closure, Mat. Vesnik 32 (4) (1980), 439-449.D. Jankovic and I. Reilly, On semi-separation properties, Indian J. Pure Appl. Math. 16 (1985), 957-964.Levine, N. (1970). Generalized closed sets in topology. Rendiconti del Circolo Matematico di Palermo, 19(1), 89-96. doi:10.1007/bf02843888Veličko, N. V. (1968). -closed topological spaces. Eleven Papers on Topology, 103-118. doi:10.1090/trans2/078/0