Transitivities of maps of generalized topological spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908706950610944 |
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| author | Zand, M. R. Ahmadi Baimani, N. |
| author_facet | Zand, M. R. Ahmadi Baimani, N. |
| contents | In this work, we present several new findings regarding the concepts of orbit-transitivity, strict orbit-transitivity, $ω$-transitivity, and $μ$-open-set transitivity for self-maps on generalized topological spaces.
Let $(X,μ)$ denote a generalized topological space. A point $x \in X$ is said to be \textit{quasi-$μ$-isolated} if there exists a $μ$-open set $U$ such that $x \in U$ and $i_μ(U \setminus c_μ(\{x\})) = \emptyset$. We prove that $x$ is a quasi-$μ$-isolated point of $X$ precisely when there exists a $μ$-dense subset $D$ of $X$ for which $x$ is a $μ_D$-isolated point of $D$. Moreover, in the case where $X$ has no quasi-$μ$-isolated points, we establish that a map $f: X \to X$ is orbit-transitive (or strictly orbit-transitive) if and only if it is $ω$-transitive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transitivities of maps of generalized topological spaces Zand, M. R. Ahmadi Baimani, N. General Topology Dynamical Systems 54A05, 37C35 In this work, we present several new findings regarding the concepts of orbit-transitivity, strict orbit-transitivity, $ω$-transitivity, and $μ$-open-set transitivity for self-maps on generalized topological spaces. Let $(X,μ)$ denote a generalized topological space. A point $x \in X$ is said to be \textit{quasi-$μ$-isolated} if there exists a $μ$-open set $U$ such that $x \in U$ and $i_μ(U \setminus c_μ(\{x\})) = \emptyset$. We prove that $x$ is a quasi-$μ$-isolated point of $X$ precisely when there exists a $μ$-dense subset $D$ of $X$ for which $x$ is a $μ_D$-isolated point of $D$. Moreover, in the case where $X$ has no quasi-$μ$-isolated points, we establish that a map $f: X \to X$ is orbit-transitive (or strictly orbit-transitive) if and only if it is $ω$-transitive. |
| title | Transitivities of maps of generalized topological spaces |
| topic | General Topology Dynamical Systems 54A05, 37C35 |
| url | https://arxiv.org/abs/2511.06241 |