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)?
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:
If in addition it also has the following property:
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 $\lim {}_{\xi} \mathcal{F} \subseteq \lim {}_{\xi} \mathcal{G}.$