Possible heights of Alexandroff square transformation groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2018
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| _version_ | 1866910300005990400 |
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| author | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Yaghmaeian, Reza Yahyaoghli, Hamed |
| author_facet | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Yaghmaeian, Reza Yahyaoghli, Hamed |
| contents | In the following text we compute possible heights of $\mathbb A$ (Alexandroff square), $\mathbb O$ (unit square $[0,1]\times[0,1]$ with lexicographic order topology) and $\mathbb U$ (unit square $[0,1]\times[0,1]$ with induced topology of Euclidean plane). We prove $P_h(\mathbb{A})=\{n:n\geq5\}\cup\{+\infty\}$, $P_h(\mathbb{O})=\{n:n\geq4\}\cup\{+\infty\}$, $P_h(\mathbb{U})=\{n:n\geq1\}\cup\{+\infty\}$ (where for topological space $X$, by $P_h(X)$ we mean the collection of heights of transformation groups with phase space $X$. In this way we also prove that there is not any topological transitive (resp. Devaney chaotic) Alexandroff square transformation group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1810_01315 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Possible heights of Alexandroff square transformation groups Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Yaghmaeian, Reza Yahyaoghli, Hamed General Topology Geometric Topology 54H15, 54H20 In the following text we compute possible heights of $\mathbb A$ (Alexandroff square), $\mathbb O$ (unit square $[0,1]\times[0,1]$ with lexicographic order topology) and $\mathbb U$ (unit square $[0,1]\times[0,1]$ with induced topology of Euclidean plane). We prove $P_h(\mathbb{A})=\{n:n\geq5\}\cup\{+\infty\}$, $P_h(\mathbb{O})=\{n:n\geq4\}\cup\{+\infty\}$, $P_h(\mathbb{U})=\{n:n\geq1\}\cup\{+\infty\}$ (where for topological space $X$, by $P_h(X)$ we mean the collection of heights of transformation groups with phase space $X$. In this way we also prove that there is not any topological transitive (resp. Devaney chaotic) Alexandroff square transformation group. |
| title | Possible heights of Alexandroff square transformation groups |
| topic | General Topology Geometric Topology 54H15, 54H20 |
| url | https://arxiv.org/abs/1810.01315 |