The answers to two problems on maximal point spaces of domains
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917936607789056 |
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| author | Xi, Xiaoyong Shen, Chong Zhao, Dongsheng |
| author_facet | Xi, Xiaoyong Shen, Chong Zhao, Dongsheng |
| contents | A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ (with the relative Scott topology). We first construct an example to show that the set of maximal points of an ideal domain $P$ need not be a $G_δ$-set in the Scott space $ΣP$, thereby answering an open problem from Martin (2003). In addition, Bennett and Lutzer (2009) asked whether $X$ and $Y$ are domain-representable if their product space $X \times Y$ is domain-representable. This problem was first solved by Önal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The answers to two problems on maximal point spaces of domains Xi, Xiaoyong Shen, Chong Zhao, Dongsheng General Topology A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ (with the relative Scott topology). We first construct an example to show that the set of maximal points of an ideal domain $P$ need not be a $G_δ$-set in the Scott space $ΣP$, thereby answering an open problem from Martin (2003). In addition, Bennett and Lutzer (2009) asked whether $X$ and $Y$ are domain-representable if their product space $X \times Y$ is domain-representable. This problem was first solved by Önal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem. |
| title | The answers to two problems on maximal point spaces of domains |
| topic | General Topology |
| url | https://arxiv.org/abs/2502.18833 |