Abstract
In this paper, we unify previous definitions of weakened open subsets in a given topological space. We felt necessary to make this unification since we observed recently too many definitions, actually, more or less significant. We also show that our new framework is more general than the known supra-topological structure. Finally we show that, in our new framework, we can define well several standard topologies like concepts such as convergence, continuity, and compactness. We expect that our new framework could find application in images processing.