Lai, Hongliang and Tholen, Walter - A Note on the Topologicity of Quantale-Valued Topological Spaces

lmcs:2640 - Logical Methods in Computer Science, August 15, 2017, Volume 13, Issue 3
A Note on the Topologicity of Quantale-Valued Topological Spaces

Authors: Lai, Hongliang and Tholen, Walter

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.

Source : oai:arXiv.org:1612.09504
DOI : 10.23638/LMCS-13(3:12)2017
Volume: Volume 13, Issue 3
Published on: August 15, 2017
Submitted on: August 15, 2017
Keywords: Computer Science - Logic in Computer Science