Partial Orders of Bijectively Related or Homeomorphic Topologies
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| Format: | Preprint |
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2024
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| _version_ | 1866929624835948544 |
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| author | Janjoš, Aleksandar Kurilić, Miloš S. |
| author_facet | Janjoš, Aleksandar Kurilić, Miloš S. |
| contents | Topologies $τ, σ\in \mathop{\mathrm{Top}}\nolimits _X$ are bijectively related, in notation $τ\sim σ$, if there are continuous bijections $f: (X, τ)\rightarrow (X, σ)$ and $g: (X, σ)\rightarrow (X, τ)$. Defining $[τ]_{\cong}=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\cong τ\}$ and $[τ]_{\sim }=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\sim τ\}$ we show that for each infinite 1-homogeneous linear order ${\mathbb L}$ there is a topology $τ\in \mathop{\mathrm{Top}}\nolimits _{|L|}$ such that: (a) $\langle [τ]_{\cong}, \subset \rangle \cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$ (the disjoint union of $2^{|L|}$-many copies of ${\mathbb L}$); so, each maximal chain in $[τ]_{\cong}$ is isomorphic to ${\mathbb L}$; (b) $\langle [τ]_{\sim}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}\widetilde{\mathbb L}$, where $\widetilde{\mathbb L}$ is the Dedekind completion of ${\mathbb L}$; thus, each maximal chain in $[τ]_{\sim}$ is isomorphic to $\widetilde{\mathbb L}$. If, in addition, the linear order ${\mathbb L}$ is Dedekind complete, then the topology $τ$ is weakly reversible, non-reversible and $\langle [τ]_{\sim}, \subset \rangle=\langle [τ]_{\cong}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial Orders of Bijectively Related or Homeomorphic Topologies Janjoš, Aleksandar Kurilić, Miloš S. General Topology 54A10, 54C05, 54C10, 06A06, 06A05 Topologies $τ, σ\in \mathop{\mathrm{Top}}\nolimits _X$ are bijectively related, in notation $τ\sim σ$, if there are continuous bijections $f: (X, τ)\rightarrow (X, σ)$ and $g: (X, σ)\rightarrow (X, τ)$. Defining $[τ]_{\cong}=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\cong τ\}$ and $[τ]_{\sim }=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\sim τ\}$ we show that for each infinite 1-homogeneous linear order ${\mathbb L}$ there is a topology $τ\in \mathop{\mathrm{Top}}\nolimits _{|L|}$ such that: (a) $\langle [τ]_{\cong}, \subset \rangle \cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$ (the disjoint union of $2^{|L|}$-many copies of ${\mathbb L}$); so, each maximal chain in $[τ]_{\cong}$ is isomorphic to ${\mathbb L}$; (b) $\langle [τ]_{\sim}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}\widetilde{\mathbb L}$, where $\widetilde{\mathbb L}$ is the Dedekind completion of ${\mathbb L}$; thus, each maximal chain in $[τ]_{\sim}$ is isomorphic to $\widetilde{\mathbb L}$. If, in addition, the linear order ${\mathbb L}$ is Dedekind complete, then the topology $τ$ is weakly reversible, non-reversible and $\langle [τ]_{\sim}, \subset \rangle=\langle [τ]_{\cong}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$. |
| title | Partial Orders of Bijectively Related or Homeomorphic Topologies |
| topic | General Topology 54A10, 54C05, 54C10, 06A06, 06A05 |
| url | https://arxiv.org/abs/2412.08319 |