Meta-Definition of Convergence So, just recently I realized that the idea of convergence is not "all encompassing"... Let me explain.
I thought that the topological definition of convergence was the most basic one in the following sense. If I had a metric space and defined convergence according to a metric $d$, then the topology induced by $d$ would match the topological notion of convergence. Similarly, if, for example, $X_n$ converged in probability to $X$, then this would be the same as $L^0$ convergence, which would imply a norm, which would imply a metric, which would imply a notion of convergence in the induced topology.
Then, I learned, for my surprise, that almost sure convergence was not topological, hence, there are quite useful and natural ideas of convergence that are not covered by topological convergence.
I was then left wondering if there is a sort of "meta-definition" of convergence. I mean, if to call something a "convergence" in the "blablabla sense", my definition would have to satisfy some properties. Hence, my question:
Is there a stronger definition of convergence that encapsulates both topological convergence, and almost sure convergence (and other similar notions)?
 A: Convergence space
The text below is copied from the Wikipedia article and somewhat edited.
Preliminaries and notation
Denote the power set of a set $X$ by $\wp(X).$  The upward closure or isotonization in $X$ of a family of subsets $\mathcal{B} \subseteq \wp(X)$ is defined as
$$
\mathcal{B}^{\uparrow X}
:= \left\{ S \subseteq X : B \subseteq S \text{ for some } B \in \mathcal{B} \, \right\}
= \bigcup_{B \in \mathcal{B}} \left\{ S : B \subseteq S \subseteq X \right\}.
$$
and similarly the downward closure of $\mathcal{B}$ is
$$
\mathcal{B}^{\downarrow} 
:= \left\{ S \subseteq B : B \in \mathcal{B} \, \right\} 
= \bigcup_{B \in \mathcal{B}} \wp(B).
$$
If $\mathcal{B}^{\uparrow X} = \mathcal{B}$ (resp. $\mathcal{B}^{\downarrow} = \mathcal{B}$) then $\mathcal{B}$ is said to be upward closed (resp. downward closed) in $X.$
For any families $\mathcal{C}$ and $\mathcal{F},$ declare that $\mathcal{C} \leq \mathcal{F}$ if and only if for every $C \in \mathcal{C},$ there exists some $F \in \mathcal{F}$ such that $F \subseteq C$
or equivalently, if $\mathcal{F} \subseteq \wp(X),$ then $\mathcal{C} \leq \mathcal{F}$ if and only if $\mathcal{C} \subseteq \mathcal{F}^{\uparrow X}.$
The relation $\leq$ defines a preorder on $\wp(\wp(X)).$ If $\mathcal{F} \geq \mathcal{C},$ which by definition means $\mathcal{C} \leq \mathcal{F},$ then $\mathcal{F}$ is said to be subordinate to $\mathcal{C}$ and also finer than $\mathcal{C},$ and $\mathcal{C}$ is said to be coarser than $\mathcal{F}.$ The relation $\,\geq\,$ is called subordination.  Two families $\mathcal{C}$ and $\mathcal{F}$ are called equivalent (with respect to subordination $\geq$) if $\mathcal{C} \leq \mathcal{F}$ and $\mathcal{F} \leq \mathcal{C}.$
A filter on a set $X$ is a non-empty subset $\mathcal{F} \subseteq \wp(X)$ that is upward closed in $X,$ closed under finite intersections, and does not have the empty set as an element (i.e. $\varnothing \not\in \mathcal{F}$).  A prefilter is any family of sets that is equivalent (with respect to subordination) to some filter or equivalently, it is any family of sets whose upward closure is a filter.  A family $\mathcal{B}$ is a prefilter, also called a filter base, if and only if $\varnothing \not\in \mathcal{B} \neq \varnothing$ and for any $B, C \in \mathcal{B},$ there exists some $A \in \mathcal{B}$ such that $A \subseteq B \cap C.$
A filter subbase is any non-empty family of sets with the finite intersection property; equivalently, it is any non-empty family $\mathcal{B}$ that is contained as a subset of some filter (or prefilter), in which case the smallest (with respect to $\subseteq$ or $\leq$) filter containing $\mathcal{B}$ is called the filter (on $X$) generated by $\mathcal{B}$.
The set of all filters (resp. prefilters, filter subbases, ultrafilters) on $X$ will be denoted by $\operatorname{Filters}(X)$ (resp. $\operatorname{Prefilters}(X),$ $\operatorname{FilterSubbases}(X),$ $\operatorname{UltraFilters}(X)$).
The principal or discrete filter on $X$ at a point $x \in X$ is the filter $\{ x \}^{\uparrow X}.$
Definition of (pre)convergence spaces
For any $\xi \subseteq X \times \wp(\wp(X)),$ if $\mathcal{F} \subseteq \wp(X)$ then define
$$\lim {}_\xi \mathcal{F} := \left\{ x \in X : \left( x, \mathcal{F} \right) \in \xi \right\}$$
and if $x \in X$ then define
$$\lim {}^{-1}_{\xi} (x) := \left\{ \mathcal{F} \subseteq \wp(X) : \left( x, \mathcal{F} \right) \in \xi \right\}$$
so if $\left( x, \mathcal{F} \right) \in X \times \wp(\wp(X))$ then $x \in \lim {}_{\xi} \mathcal{F}$ if and only if $\left( x, \mathcal{F} \right) \in \xi.$  The set $X$ is called the underlying set of $\xi$ and is denoted by $\left| \xi \right| := X.$
A preconvergence on a non-empty set $X$ is a binary relation $\xi \subseteq X \times \operatorname{Filters}(X)$ with the following property:

*

*Isotone: if $\mathcal{F}, \mathcal{G} \in \operatorname{Filters}(X)$ then $\mathcal{F} \leq \mathcal{G}$ implies $\lim {}_{\xi} \mathcal{F} \subseteq \lim {}_{\xi} \mathcal{G}$.

If in addition it also has the following property:


*Centered: if $x \in X$ then $x \in \lim {}_{\xi} \left( \{ x \}^{\uparrow X} \right)$.

then the preconvergence $\xi$ is called a convergence on $X.$
A generalized convergence or a convergence space (resp. a preconvergence space) is a pair consisting of a set $X$ together with a convergence (resp. preconvergence) on $X.$
A preconvergence $\xi \subseteq X \times \operatorname{Filters}(X)$ can be canonically extended to a relation on $X \times \operatorname{Prefilters}(X),$ also denoted by $\xi,$ by defining
$$\lim {}_{\xi} \mathcal{F} := \lim {}_{\xi} \left( \mathcal{F}^{\uparrow X} \right)$$
for all $\mathcal{F} \in \operatorname{Prefilters}(X).$ This extended preconvergence will be isotone on $\operatorname{Prefilters}(X),$ meaning that if $\mathcal{F}, \mathcal{G} \in \operatorname{Prefilters}(X)$ then $\mathcal{F} \leq \mathcal{G}$ implies $\lim {}_{\xi} \mathcal{F} \subseteq \lim {}_{\xi} \mathcal{G}.$
