Extracting some supra topologies from the topology of a topological space using stacks

Document Type : Original Article


Department of Mathematics, Payame Noor University, P.O. Box, 19395-3697, Tehran, Iran


A collection $\mu$ of subsets of a nonempty set $X$ is a supra topology on $X$ whenever $\emptyset$ and $X$ belong to $\mu$, and also $\mu$ is closed under arbitrary unions. Also, a nonempty collection $\mathcal{S}$ of nonempty subsets of a nonempty set $X$ is called a stack on $X$ whenever it is closed under operation superset. In this paper, we are going to introduce an approach to extract some supra topologies from the topology of a topological space. For this purpose, we consider a topological space $(X, \tau)$ with a closed set $P$ of its subsets. Using a stack $\mathcal{S}$ on the space $(X, \tau)$ and the closure operator $cl$ associated with $\tau$, we define a supra closure operator $\lambda_P$ on $X$ to create the desired supra topology. We then characterize the form of this resulting supra topology and also determine its relationship to the initial topology of the space.


Main Subjects

1] T. M. Al-SHAMI, Some results related to supra topological spaces, J. Adv. Stud. TOPOL., 7 (2016), pp. 283–294.
[2]___ , On supra semi open sets and some applications on topological spaces, J. Adv. Stud. TOPOL., 8 (2017), pp. 144–153.
[3] A. CSA´SZA´R ,  Generalized topology, generalized continuity, Acta Math. Hungar., 96 (2002), pp. 351–357.
[4] R. Devi, S. SAMPATHKUMAR, and M. Caldas, On α-open sets and α-continuous maps, General Mathematics, 16 (2008), pp. 77–84.
[5] M. El-SHAFEI, M. Abo-ELHAMAYEL, and T. Al-SHAMI, On supra r-open sets and some applications on topological spaces, Journal of Progressive Research in Mathematics, 8 (2016), pp. 1237–1248.
[6] S. JAFARI and S. TAHILIANI, Supra β-open sets and supra β-continuity on topological spaces, Universitatis Scientiarum Budapestinensis de Rolando Eotvos nominatae, (2013), p. 61.
[7] A. S. Mashhour, A. A. ALLAM, F. S. Mahmoud, and F. H. KHEDR, On supratopological spaces, Indian J. Pure Appl. Math., 14 (1983), pp. 502–510.
[8] O. Sayed, Supra pre-open sets and supra pre-continuity on topological spaces, Scientific Studies and Research, 20 (2010).
[9] O. R. Sayed and T. NOIRI, On supra b-open sets and supra b-continuity on topological spaces, Eur. J. Pure Appl. Math., 3 (2010), pp. 295–302.
[10] A. TALABEIGI, Embedding topological spaces in a type of generalized topological spaces, Khayyam J. Math., 6 (2020), pp. 250–256.
[11]___ , Extension of topological derived set operator and topological closure set operator via a class of sets to construct generalized topologies, Caspian Journal of Mathematical Sciences (CJMS) peer, (2020).