Models of an Abstract Elementary Class as a Generalized Polish Space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912299030151168 |
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| author | Marangelis, Georgios |
| author_facet | Marangelis, Georgios |
| contents | In first order logic, it is known that you can define a topology so that the countable models of some theory $T$ form a Polish Space (i.e. completely metrizable second countable space).
In this paper we use the Baldwin- Boney Relational Presentation Theorem (from [3]; cf. 2.3) to generalize this result to the models of an Abstract Elementary Class (AEC). More specifically, we define a topology on the models of an AEC of size $λ\geq κ$, where $κ$ is the L$\ddot{o}$wenheim-Skolem number and $λ$ has to satisfy a set-theoretic assumption (see Section 4) and prove that these models form a Generalized Polish Space (i.e. a generalization of Polish Spaces i.e. completely $G$-metrizable space with weight $\leq κ$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Models of an Abstract Elementary Class as a Generalized Polish Space Marangelis, Georgios Logic Primary 03C48, 03E15, Secondary 03C75, 54E52, 54H05 In first order logic, it is known that you can define a topology so that the countable models of some theory $T$ form a Polish Space (i.e. completely metrizable second countable space). In this paper we use the Baldwin- Boney Relational Presentation Theorem (from [3]; cf. 2.3) to generalize this result to the models of an Abstract Elementary Class (AEC). More specifically, we define a topology on the models of an AEC of size $λ\geq κ$, where $κ$ is the L$\ddot{o}$wenheim-Skolem number and $λ$ has to satisfy a set-theoretic assumption (see Section 4) and prove that these models form a Generalized Polish Space (i.e. a generalization of Polish Spaces i.e. completely $G$-metrizable space with weight $\leq κ$). |
| title | Models of an Abstract Elementary Class as a Generalized Polish Space |
| topic | Logic Primary 03C48, 03E15, Secondary 03C75, 54E52, 54H05 |
| url | https://arxiv.org/abs/2503.22287 |