Defining a topology with ultrafilters Suppose $X$ is a set and let $U(X)$ denote the set of ultrafilters on $X$. Now let $a:U(X)\rightarrow X$ be a relation. Now we can define a topology $\tau$ on $X$ using this relation in the following way:
$U\in \tau$ $\iff$ for each $x\in U$, $(\mathcal{F},x)\in a \implies U\in \mathcal{F}$ (note here $\mathcal{F}\in U(X)$).
Question: Is it true that an ultrafilter $\mathcal{F}$ converges to a points $x\in X$ i.e. $\mathcal{F}\rightarrow x$ $\iff$$(\mathcal{F},x)\in a$?
It is quite easy to see that $(\mathcal{F},x)\in a\implies \mathcal{F}\rightarrow x$, however I am not sure if the other implication: $\mathcal{F}\rightarrow x \implies (\mathcal{F},x)\in a$, is true.
Note: When I say $\mathcal{F}\rightarrow x$, I mean that every $U\in \tau$ with $x\in U$ is in $\mathcal{F}$ (i.e. ultrafilter ocnvergence in the topological sense).
 A: No, this is not true in general.  For a very simple example, if $F\in U(X)$ is a principal ultrafilter associated to a point $x\in X$, then $F$ always converges to $x$, even if the relation $a$ does not say so.  More subtly, convergence of ultrafilters with respect to a topology always satisfies a certain "continuity" condition.  For instance, if $X$ is an infinite set and $a$ says that every ultrafilter converges to every point except for a single nonprincipal ultrafilter $F$ which does not converge at all, then $\tau$ will still be the indiscrete topology and so $F$ will converge to every point with respect to $\tau$.  Intuitively, the problem is that $F$ can be "approximated" by other ultrafilters, and so its convergence properties must be compatible with theirs.
The precise restrictions on what $a$ must satisfy for your question to have an affirmative answer is that it must be the structure map of a "relational $\beta$-module".  See https://ncatlab.org/nlab/show/relational+beta-module for the details.  See also my answer at Which "limit of ultrafilter" functions induce a compact Hausdorff topological structure? for an explanation of how the second restriction on $a$ can be thought of as "continuity", at least in the case that $a$ is a function.
A: Denote by $\mathcal{F}_p$ the ultrafilter fixed on $p$.   
No , this is not true, take $X = \mathbb{Z}$, and define the convergence as $(\mathcal{F},x) \in a$ iff $\{x-1,x, x+1\} \in \mathcal{F}$. (This also obeys the common axiom $\forall x \in X: (\mathcal{F}_x,x) \in a$)
Then the definition of openness: 
Let $O$ be open and suppose $x \in O$. Note that $(\mathcal{F}_{x-1}, x) \in a$ so that $O \in \mathcal{F}_{x-1}$ or $x-1 \in O$. 
Similarly $x+1 \in O$ as well. Now by two sided induction we see that $\mathbb{Z} \subseteq O$.
So the only open sets are $\emptyset$ and $X$, i.e. this convergence induces the indiscrete topology on $X$. But then $\mathcal{F}_0 \to 2$ in the indiscrete topology, while $(\mathcal{F}_0,2) \notin a$ as $\{1,2,3\} \notin \mathcal{F}_0$.
(This is a standard example of a so-called pretopological space that is not a topological space.)
A: in this paper by Hofman and Tholen they consider all relations $a \subseteq UX \times X$, where $UX$ is the set of ultrafilters on $X$. We write $\mathcal{U} \overset{a}\to x$ for $(\mathcal{U},x) \in a$. We also consider all $\tau \subseteq \mathscr{P}(X)$ (like topologies are).
There are two natural maps : first $\psi$ from "subfamilies" to "convergences":
$$(\mathcal{U},x) \in \psi(\tau) \text{ iff } \forall A \in \tau: (x \in A \implies A \in \mathcal{U})$$
(inspired by: a filter converges to $x$ iff it contains all neighbourhoods of $x$).
and also a map in the reverse direction $\phi$, mapping a "convergence" $a$ to a "topology":
$$ A \in \mathcal \phi(a) \text{ iff } \forall (\mathcal{U},x) \in a: (x \in A \implies A \in \mathcal{U})$$
inspired by: a set is open, iff every point in it is a neighbourhood of it, where a neighbourhood of $x$ means that the set lies in all filters converging to $x$.
Theorem A in the quoted paper says that $\tau \subset \mathscr{P}(X)$ is a topology iff $\tau = \phi(\psi(\tau))$.
I find this rather elegant, but I don't know if it qaulifies as a characterisation of topologies via convergences.
