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For questions about frames (=lattices with arbitrary joins and such that finite meets are distributive over those, such as the open sets of a topological set) and locales (the dual category to that of frames, seen as a generalization of topological spaces), as well as about pointfree (“pointless”) topology. This includes questions both of an algebraic nature about frames and of a topological nature about locales.
6
votes
1
answer
301
views
What is the power of the “anti-halting” oracle?
Let me first ask the question, and then, as it may seem a bit cryptic, explain how it comes up (and whence the “anti-halting oracle” in the title):
Notations: we write $\langle m,n\rangle$ for a stand …
10
votes
0
answers
144
views
Closed sets versus closed sublocales in general topology in constructive math
This question is set in constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF.
Short version of the question: if $X$ is a sober top …
7
votes
1
answer
235
views
Dissolution of a topos
The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a fra …
5
votes
1
answer
80
views
Preimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs dir …
3
votes
0
answers
98
views
Constructively valid reference for the soberness of discrete spaces and points of a locale c...
I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of …
3
votes
1
answer
217
views
Computing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“ …
2
votes
0
answers
98
views
What is known about sublocales defined by regular nuclei?
(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.)
I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements …