Irreducible Maps and Isomorphisms of Boolean Algebras of Regular Open Sets and Regular Ideals
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913886180999168 |
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| author | Pitts, David R. |
| author_facet | Pitts, David R. |
| contents | Let $π: Y\rightarrow X$ be a continuous surjection between compact Hausdorff spaces $Y$ and $X$ which is irreducible in the sense that if $F\subsetneq Y$ is closed, then $π(F)\neq X$. We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of $X$, the regular open sets of $Y$, the regular ideals of $C(X)$ and the regular ideals of $C(Y)$.
We call $X$ and $Y$ Boolean equivalent if the regular open sets of $X$ and the regular open sets of $Y$ are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00598 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Irreducible Maps and Isomorphisms of Boolean Algebras of Regular Open Sets and Regular Ideals Pitts, David R. General Topology Operator Algebras 54H99, 46J10 (Primary) 06E15, 54G05 (Secondary) Let $π: Y\rightarrow X$ be a continuous surjection between compact Hausdorff spaces $Y$ and $X$ which is irreducible in the sense that if $F\subsetneq Y$ is closed, then $π(F)\neq X$. We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of $X$, the regular open sets of $Y$, the regular ideals of $C(X)$ and the regular ideals of $C(Y)$. We call $X$ and $Y$ Boolean equivalent if the regular open sets of $X$ and the regular open sets of $Y$ are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces. |
| title | Irreducible Maps and Isomorphisms of Boolean Algebras of Regular Open Sets and Regular Ideals |
| topic | General Topology Operator Algebras 54H99, 46J10 (Primary) 06E15, 54G05 (Secondary) |
| url | https://arxiv.org/abs/2401.00598 |