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What are the bounded lattices $\mathcal{L}=(Q,\lor,\land,\bot,\top)$ called that are isomorphic to topological spaces? I.e. those such that there exists a topological space $(X,\tau)$ and a bijection $f:Q\to\tau$ such that $f(\top)=X$ and $f(\bot)=\emptyset$ with $f(\bigvee_{i\in I}x_i)=\bigcup_{i\in I}f(x_i)$ and $f(a\land b)=f(a)\cap f(b)$?

It would seem the study of these lattices is essentially just the study of point-set topology, thus I imagine they must have a name? What are they called?

user3865123
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1 Answers1

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They are called spatial frames. They are, in particular, complete Heyting algebras. Their formal duals are called spatial locales.

Without the "spatial" condition you get a slightly more general class of objects called frames and locales which are more natural and better-behaved in some respects, and their study is sometimes called pointless topology.

Qiaochu Yuan
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  • Thank you. I am curious why is there such a big focus in differentiating them? Is it just because its easier to study them in terms of topological spaces and its trivial to translate over all the ideas to spatial frames? – user3865123 Sep 07 '20 at 00:00
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    @user3865123: among other things, you can define a locale internal to a topos (https://ncatlab.org/nlab/show/locale#CategoryOfLocales) and the result is better behaved than trying to define topological spaces internal to a topos. In a topos it becomes more true that a locale may not have "enough points" to be spatial, analogous to how varieties can behave over non-algebraically closed fields. – Qiaochu Yuan Sep 07 '20 at 00:03
  • Lol I understand very little of what you just said. Though I hope one day to be as educated as you on these matters. – user3865123 Sep 07 '20 at 00:05