Transitivities of maps of generalized topological spaces

Abstract

In this work, we present several new findings regarding the concepts of orbit-transitivity, strict orbit-transitivity, $\omega$-transitivity, and $\mu$-open-set transitivity for self-maps on generalized topological spaces. Let $(X,\mu)$ denote a generalized topological space. A point $x \in X$ is said to be \textit{quasi-$\mu$-isolated} if there exists a $\mu$-open set $U$ such that $x \in U$ and $i_\mu(U \setminus c_\mu(\{x\})) = \emptyset$. We prove that $x$ is a quasi-$\mu$-isolated point of $X$ precisely when there exists a $\mu$-dense subset $D$ of $X$ for which $x$ is a $\mu_D$-isolated point of $D$. Moreover, in the case where $X$ has no quasi-$\mu$-isolated points, we establish that a map $f: X \to X$ is orbit-transitive (or strictly orbit-transitive) if and only if it is $\omega$-transitive.