On Maximal \(\mu ^*\) -Open Set, Minimal \(\mu ^*\) -Closed Set, Mean \(\mu ^*\) -Open Set and Its Applications in Discrete Dynamical Systems

The main aim of this article is to introduce the weak form of open set known as maximal \(\mu ^*\) -open set in generalized topology in presence of topology and discuss some of its properties and it is shown that this concept is independent from the existing concept of maximal open set and maximal \(\mu \) -open set. The relation between \(\mu ^*\) -interior and \(\mu ^*\) -closure of maximal \(\mu ^*\) -open set are obtained. Also for a proper nonnull \(\mu ^*\) -open set P in a space \((X, \mu X, \tau )\) such that \((X-P)\) is finite then \(\exists \) maximal \(\mu ^*\) -open set Q such that \(P \subseteq Q\) . These concepts can be used as a tool in other branch of mathematics like graph theory to compare different class of edge sets as generalized topology can be formed from the edge set of maximal path of directed graph and also to characterize maximal invariant set of nonlinear discrete time control dynamical systems.