# Questions tagged [locales]

For questions about locales, a generalization of topological spaces which need not have points. Their study is called pointless topology.

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### Proving that the pointful and the pointless forms of the well-inside relation coincide

In Stone Spaces by Johnstone, a pointless version of the well-inside order is given: Definition 1. Let $L$ be a locale and let $x, y \in L$. $x$ is said to be well-inside $y$ iff \begin{equation*} x ...
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### What is the universal property of the prime spectrum of a commutative rig?

Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, ...
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### Info on the locale of surjections from the Natural Numbers to the Real Numbers

On the nlab page for locales, it states that there is locale for the surjections from the Naturals to the Reals. This locale has no points (i.e. elements), since there are no such surjections, but the ...
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### How to represent a collection of closed sets in a locale of opens?

Given a collection of closed sets $\mathcal{F}$ of a topological space $X$, how do I relate this with a locale $\mathcal{O}(X)$ of opens of $X$?. The obvious case is when we have a collection of ...
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### Regular locale is spatial

Guided by topology I'm expecting this to be true, but I cannot find a proof for the following statement: Every regular locale is spatial.
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### Stone-Äech compactification for locales

I'm reading about the Stone-Äech compactification for locales, which (I think) states that for every locale $L$ there exists a compact regular locale $\kappa L$ and a morphism $r : L \to \kappa L$ ...
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### What are are some illustrative (non-)examples of proper morphisms?

"Proper" is an adjective used to describe a morphism of spacesātopological spaces, schemes, locales, etcāthat is sufficiently nice and has some neat properties. Between topological spaces a morphism ...
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### Rings and locales

Let $A$ be a ring. Given a monoid (=multiplicative system) $S \subseteq A$ one can define the localization of $A$ at $S$ as the $A$-algebra $$j_S \colon A \to A[S^{-1}]$$ which is universal among the ...