| Autor: |
Hongliang Lai, Walter Tholen |
| Jazyk: |
angličtina |
| Rok vydání: |
2017 |
| Předmět: |
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| Zdroj: |
Logical Methods in Computer Science, Vol Volume 13, Issue 3 (2017) |
| Druh dokumentu: |
article |
| ISSN: |
1860-5974 |
| DOI: |
10.23638/LMCS-13(3:12)2017 |
| Popis: |
For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for ${\sf{V}}$ completely distributive, ${\sf{V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As a consequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paper we give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be a spatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yields complete distributivity of ${\sf{V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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