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edited Dec 4, 2018 at 17:36

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Edit: I'm trying to address Gerhard Paseman comment:

Let me start with this space of "generic real number" $G \subset \mathbb{R}$ defined as the intersection of all subspaces of density $1$ that I mentioned at the end. (Note to the experts:I am always confused on whether this space is already boolean or not, I'm assuming it is, but if it is not please tell me. If it is not then some of the things that I say below might require to move to a certain Boolean cover of $G$)

It has some quite nice properties that are easy to explain: For example bounded continuous functions from ${G}$ to $\mathbb{C}$ are exactly the same as element of $L^{\infty}(\mathbb{R},\mathbb{C})$.

The fact that $G$ has no points simply corresponds to the fact that you can't evaluate a function in $L^{\infty}$ (in localic terms: these are functions defined only on generic reals)

more generally things defined over it are things defined "almost everywhere" and that for example one can define what are bundle of Hilbert spaces over $G$ and under countability condition they are exactly the same as measurable fields of Hilbert spaces on $\mathbb{R}$ with morphisms between them being already defined up to equality almost everywhere.

But the notion of bundle of Hilbert spaces over $G$ makes sense without countability conditions and is considerably more robust than the notion of measurable fields of spaces when we drop the countability assumption (it is equivalent to the notion of $W^*$-modules over $L^{\infty}(\mathbb{R},\mathbb{C})$ ).

But of course, all those require first to show that $G$ is non-empty.

An important applications of $S$ is for forcing in set theory. Though forcing existed before this kind of consideration this gives a very nice geometric picture of what forcing is (and this is completely equivalent to the usual formulation of forcing, its just a different language).

Essentially, by replacing "sets" with "sheaves over the Space $S$" (or more precisely, over some boolean cover of $S$, or looking at double negation sheaves) one obtains a new model of set theory in which $I$ is actually countable. This is what Set theorists call a forcing extension that collapse $I$ and $\omega$.

informally (though it can be made formal with things like internal logic) its because working with sheaves over $S$, means working with structure which depends on a parameter $s \in S$, and informally such a parameter is the data of a surjection $\mathbb{N} \rightarrow I$

Other kind of forcing corresponds to other space similar to $S$, and one generally proves they are non-empty by the same argument.

The fact that space $S$ is non-empty is really equivalent to the fact that one can collapse cardinals by forcing, so it is a bit hard to find application outside of logic. But I'll see if I can think of something nice enough to be explained here.

Edit: I'm trying to address Gerhard Paseman comment:

It has some quite nice properties that are easy to explain: For example bounded continuous functions from ${G}$ to $\mathbb{C}$ are exactly the same as element of $L^{\infty}(\mathbb{R},\mathbb{C})$.

The fact that $G$ has no points simply corresponds to the fact that you can't evaluate a function in $L^{\infty}$ (in localic terms: these are functions defined only on generic reals)

But of course, all those require first to show that $G$ is non-empty.

Other kind of forcing corresponds to other space similar to $S$, and one generally proves they are non-empty by the same argument.

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created Dec 4, 2018 at 13:22

Simon Henry

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I like to call this result the localic Baire category theorem, and it plays essentially the same role as Baire category theorem: it lets you "construct" object by showing that some spaces are non-empty because they are dense. Of course because we are working with locales one cannot necessarily "construct" an object at the end, because a non-empty locale does not neccessary have a points. But often having a non-empty space of "Something" is as good as having a non-explicit construction of a "Something".

Note that the usual Bair category theorem can be seen as a mixture of this localic Baire category theorem and some classical results saying that locales satisfying some properties have enough points. And if you stare at the usual proof of Baire category theorem you will see that one can clearly recognize the proof of these statement of existence of points)

So in addition to all the usual application of Baire category theorem, Here is the simplest and most famous example of this:

Let $I$ be an infinite set (preferably uncountable).

Consider the space $I^{\mathbb{N}}$ to the space of sequences with values in $I$ (with the product topology). For each $i \in I$, consider $D_i \subset I^{\mathbb{N}}$ the subset of sequence which take the value $i \in I$ at least once. One easily see that each $D_i$ is a dense open subset of $I^{\mathbb{N}}$.

Consequence: $$ S:=\left( \bigcap_{i\in I} D_i \right) \neq \emptyset $$

Of course the intersection means "intersection of sublocales" and by non-empty I mean that the resulting sublocale is not the empty sublocales. And it is non-empty because $\overline{S} = I^{\mathbb{N}}$.

This $S$ is essentially by definition "The space of surjections $\mathbb{N} \twoheadrightarrow I$ (indeed $f \in D_i$ means that $i$ is in the image of $f$, so $f \in S$, means that all $i \in I$ are in the image.)

So for any uncountable set, one has a non empty space of surjection from $\mathbb{N}$ to $I$. Of course this space cannot have any point if $I$ is not countable.

I like to sum up this by the fact that in locales theory "every discrete space is (geometrically) countable", but this more of a slogan.

More concretely this is directly related to the fact that any set can be made countable in a forcing extension of set theory, which is the key idea that lead to the proof By cohen of the independance of the continuum hypothesis. (though of course forcing and Cohen's proof existed before one knew about the connection with this geometric point of view, but it is still true that these proofs are essentially the same as these observations, a discussion of Cohen's proof in this language can be found in Moerdijk&MacLane sheaves in geometry and logic).

Another example is the fact that the space (read locale) of "Lebesgue Generic real number" is dense in the space of real number. Indeed Lebesgue generic real number are the things in the intersection of all sub-locales of density $1$ in $\mathbb{R}$, and these are all individually dense.