Locales as spaces in outer models
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913575297089536 |
|---|---|
| author | Bannister, Nathaniel |
| author_facet | Bannister, Nathaniel |
| contents | Let M be a transitive model of set theory and X be a space in the sense of M. Is there a reasonable way to interpret X as a space in V? A general theory due to Zapletal provides a natural candidate which behaves well on sufficiently complete spaces (for instance Čech complete spaces) but behaves poorly on more general spaces - for instance, the Zapletal interpretation does not commute with products. We extend Zapletal's framework to instead interpret locales, a generalization of topological spaces which focuses on the structure of open sets. Our extension has a number of desirable properties; for instance, localic products always interpret as spatial products. We show that a number of localic notions coincide exactly with properties of their interpretations; for instance, we show a locale is $T_U$ if and only if all its interpretations are $T_1$, a locale is $I$-Hausdorff if and only if all its interpretations are $T_2$, a locale is regular if and only if all its interpretations are $T_3$, and a locale is compact if and only if all its interpretations are compact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05967 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locales as spaces in outer models Bannister, Nathaniel Logic Let M be a transitive model of set theory and X be a space in the sense of M. Is there a reasonable way to interpret X as a space in V? A general theory due to Zapletal provides a natural candidate which behaves well on sufficiently complete spaces (for instance Čech complete spaces) but behaves poorly on more general spaces - for instance, the Zapletal interpretation does not commute with products. We extend Zapletal's framework to instead interpret locales, a generalization of topological spaces which focuses on the structure of open sets. Our extension has a number of desirable properties; for instance, localic products always interpret as spatial products. We show that a number of localic notions coincide exactly with properties of their interpretations; for instance, we show a locale is $T_U$ if and only if all its interpretations are $T_1$, a locale is $I$-Hausdorff if and only if all its interpretations are $T_2$, a locale is regular if and only if all its interpretations are $T_3$, and a locale is compact if and only if all its interpretations are compact. |
| title | Locales as spaces in outer models |
| topic | Logic |
| url | https://arxiv.org/abs/2411.05967 |