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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 6th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Definition The __dissolution locale__ $\mathfrak{C}L$ of a [[locale]] $L$ is defined as the poset of its [[sublocales]] (equivalently: [[nuclei]] on $L$) equipped with the relation of reverse inclusion. There is a canonical morphism of locales $$\iota\colon\mathfrak{C}L \to L$$ such that the map $\iota^*$ sends an open $a\in L$ to the open in $\mathfrak{C}L$ given by the open sublocale of $a$. ## Interpretation The map $\mathfrak{C}L\to L$ can be considered an analogue of the canonical map $T_d \to T$ for a [[topological space]] $T$, where $T_d$ is the underlying set of $T$ equipped with the [[discrete topology]]. In particular, *discontinuous maps* $L\to M$ could be defined as morphisms of locales $\mathfrak{C}L\to M$, see Picado–Pultr, XIV.7.3. ## References Original reference: * [[John R. Isbell]], On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. [doi](https://doi.org/10.1016/0166-8641(91)90058-T). Expository account: * [[Frames and Locales]], see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by $\mathcal{Sl}(L)^{op}$ (III.3.2) or by $\mathfrak{C}(L)$ (III.5.2) and the dissolution locale is denoted by $\mathfrak{S}(L)$ (XIV.7.2). <a href="https://ncatlab.org/nlab/revision/dissolution+locale/1">v1</a>, <a href="https://ncatlab.org/nlab/show/dissolution+locale">current</a>

   Created:

   Definition

   The dissolution locale $\mathfrak{C}L$ of a locale $L$ is defined as the poset of its sublocales (equivalently: nuclei on $L$) equipped with the relation of reverse inclusion.

   There is a canonical morphism of locales

   $$\iota\colon\mathfrak{C}L \to L$$

   such that the map $\iota^*$ sends an open $a\in L$ to the open in $\mathfrak{C}L$ given by the open sublocale of $a$.

   Interpretation

   The map $\mathfrak{C}L\to L$ can be considered an analogue of the canonical map $T_d \to T$ for a topological space $T$, where $T_d$ is the underlying set of $T$ equipped with the discrete topology.

   In particular, discontinuous maps $L\to M$ could be defined as morphisms of locales $\mathfrak{C}L\to M$, see Picado–Pultr, XIV.7.3.

   References

   Original reference:

   • John R. Isbell, On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. doi.

   Expository account:

   • Frames and Locales, see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by $\mathcal{Sl}(L)^{op}$ (III.3.2) or by $\mathfrak{C}(L)$ (III.5.2) and the dissolution locale is denoted by $\mathfrak{S}(L)$ (XIV.7.2).

   v1, current