The recursively enumerable sets are a collection of subsets of $\mathbb{N}$, whose definition is well-known and can be found on Wikipedia here. Yesterday, i happened to stumbled upon a definition of "Generalized Topological Spaces", here (definition 2.2.1) (henceforth referred to as GTS). The definition is extensive, and i ask the reader to check the link, but for the sake of the question text; a triple $(X, Op_X, Cov_X)$, with a set $X$, a collection of open sets $Op_X\in 2^X$, and admissible coverings $Cov_X\in 2^{2^X}$ (this last one sets GTS apart from regular topology; unions are not arbitrary but instead restricted to $Cov_X$) forms a GTS if the triple satisfies some conditions, A1 to A8.
We can then check if the triple $(\mathbb{N}, RE, Cov_{RE})$ forms such a space (where $RE$ is the collection of recursively enumerable sets, and $Cov_{RE}$ is the collection of collections $C$ of $RE$ elements such that $C$ is itself recursively enumerable[1]). It turns out that it doesn't: conditions A7 and A8 (the saturation[2] and regularity axioms) fail for this triple.
The next step here is to consider what happens if we simply ignore said failing conditions, or, in other words, to further generalize the GTS. The same text that presents the definition of GTS explains that said definition is related to Grothendieck topologies, but here we hit a snag; while the definition of GTS was explained with plain set-theoretical language, Grothendieck topology is, as far as i can tell, a concept deeply rooted in Category Theory, whose language i am still far from understanding. Nevertheless, one can navigate the ncatlab and reach the definition of Site, here, which is a category together with a Coverage, something defined here. My understanding is that coverage is the most general definition in this context, and that one obtains Grothendieck (pre)topologies by imposing additional conditions on a coverage (i am not sure where exactly do GTS fit in all this, but i believe sites are indeed a generalization of GTS).
The actual question i ask here breaks into multiple parts:
- Am i right about what a site is? That is, if we "de-categorify" the definition of site (and coverage too, of course), do we end up with something like the definition of GTS, except with less conditions?
- If so, does the triple $(\mathbb{N}, RE, Cov_{RE})$ form a site? That is, is $Cov_{RE}$ indeed a coverage for $\mathbb{N}$? For example, is it "stable under pullback" (whatever that means!)?
- If that is true as well, which additional "saturation conditions" (see here) does $Cov_{RE}$ satisfy? I imagine that, not enough for it to be a proper Grothendieck topology, but maybe enough for a pretopology?
[1] – A slight abuse of language is carried out when it is said that "$C$ is recursively enumerable" (one would expect $C\in RE$ from this sentence alone, but actually $C\in 2^{RE}$ in this specific case); for those unconfortable with it, an equivalent way to define $Cov_{RE}$ is as follows. First, fix $\phi : \mathbb{N} \rightarrow RE$, a computable enumeration of RE itself. Then $Cov_{RE}$ is $\{C \in 2^{RE} \mid \exists S \in RE , \forall s \in RE, s \in C \leftrightarrow \exists n\in S, s = \phi(n) \}$, that is, a collection $C$ of RE elements belongs to $Cov_{RE}$ iff there exists $S\in RE$ such that one can map $\phi$ over $S$ and obtain $C$ as the result.
[2] – Note that the "saturation axiom" here is a specific one for GTS, the category-theory related definitions further in the question have their own, multiple, saturation conditions.
Best Answer
Let us suppose we are dealing with an arbitrary partially ordered set $(P, \leq)$. In the particular case of topological spaces, $P$ is some collection of subsets of $X$, the underlying space. We can consider $P$ as a category in a canonical way as follows: the set of objects is $P$, there is at most one arrow between each $x, y \in P$, and there is an arrow between $x$ and $y$ iff $x \leq y$.
A sieve on an object $x$ can be defined as a collection $S \subseteq \{(f, z) | f : z \to x\}$ which satisfies the property that for every $(f, z) \in S$ and every $g : w \to z$, we have $(f \circ g, w) \in S$.
When we are talking about a partially ordered set, the first component of $(f, z)$ where $f : z \to x$ doesn't add any information (other than the fact that $z \leq x$) since there is at most one $f : z \to x$. Thus, we can equivalently consider a sieve $S$ on $x$ to be a collection $S \subseteq \{z \in P : z \leq x\}$ s.t. for all $z \in S$, for all $w \leq z$, $w \in S$. This is what I will call a PO-sieve.
Given a sieve $S$ on $y$ and an arrow $f : x \to y$, we may define the $f^*(S) = \{(g, z)| g : z \to x$ and $f \circ g \in S\}$, a sieve on $y$.
Correspondingly, given a PO-sieve $S$ on $y$ and some $x \leq y$, we may define $S_x = \{z : z \leq x$ and $z \in S\}$, a sieve on $x$.
A Grothendieck Topology on a category $C$ is a mapping from each object $x \in C$ to a family $F_x$ of sieves on $x$ which satisfies several axioms.
Correspondingly, a PO-Grothendieck Topology on a poset $P$ must be a mapping from each element $x \in P$ to a family $F_x$ of PO-sieves which satisfies the corresponding axioms.
Axiom 1 of Grothendieck Topology: for every $x \in C$, we have $\{(f, z) : f : z \to x\} \in F_x$.
Corresponding Axiom 1 of PO-Grothendieck Topology: for every $x \in P$, we have $\{z : z \leq x\} \in F_x$.
Axiom 2 of Grothendieck Topology: for every $f : x \to y$, for every sieve $S \in F_y$, we have $f^*(S) \in F_x$.
Corresponding Axiom 2 of PO-Grothendieck Topology: for every $x \leq y$ and for every PO-sieve $S \in F_y$, we have $S_x \in F_x$.
Axiom 3 of Grothendieck Topology: suppose we have $S \in F_x$. And suppose we have a sieve $P$ on $x$ such that for all $(f, z) \in S$, $f^*(P) \in F_z$. Then $P \in F_x$.
Corresponding Axiom 3 of PO-Grothendieck Topology: suppose we have $S \in F_x$. And suppose we have a PO-sieve $P$ on $x$ s.t. for all $z \in S$, $P_z \in F_z$. Then $P \in F_x$.
How does this relate to Generalised Topological Spaces? Suppose given such a generalised space. The partially ordered set $P$ is the set of opens ordered by $\subseteq$. Suppose given some collection $C$ of open sets. Define $f(C) = \{U $ open$: \exists V \in C (U \subseteq V)\}$. Note that for every such $C$, $f(C)$ is a PO-sieve. Then given $U$ open, we may define $F_U = \{f(C) : C \in cov_X$ and $\bigcup\limits_{V \in C} V = U\}$.
Let us verify that this gives us a PO-Grothendieck topology.
Axiom 1: this follows from the fact that $\{U\} \in cov_X$ for all $U$ - that is, it follows from axiom A3.
Axiom 2: this follows from axiom A5.
Axiom 3: this follows from axiom A6.
Finally, we turn to your example of $\mathbb{N}$ with "opens" recursively enumerable sets and "coverings" recursive enumerations of recursively enumerable sets. Since this satisfies axioms A3, A5, and A6, it does form a PO-Grothendieck topology.