Transitivities of maps of generalized topological spaces

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 $\mathit{\text{quasi-}}\mu \mathit{\text{-isolated}}$ if there exists a $\mu$-open set $U$ such that $x\in U$ and $i_{\mu }(U\setminus c_{\mu }(\{x\}))=\mathrm{\varnothing }$. 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.