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

Document Type : Original Article

Author

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

Abstract

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.

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