Examples of "non-topological" convergence I'm looking for some interesting examples of convergence spaces (or related notions), which use a convergence that doesn't come from a topology. 
I know about "convergence-almost-everywhere". What other examples are there?

EDIT: It seems my question is more or less a duplicate of this question on mathoverflow.

In "Handbook of Analysis and Its Foundations" one can find a vast (and arguably interesting and useful!) generalization of the "convergence almost everywhere":
Let $(P,\leq)$ be a complete lattice (this works for general posets too, but the definitions are considerably more cumbersome):
Given a net $(x_n)_{n\in A}$ in $P$ define:
$$\liminf_{n\in A} x_n = \bigvee_{n\in A} \bigwedge_{m\geq n} x_m,\,\limsup_{n\in A} x_n = \bigwedge_{n\in A} \bigvee_{m\geq n} x_m$$
We say $(x_n)_{n\to A}$ order converges to $\tilde x$, if $\liminf_{x\in A} x_n = \tilde x = \limsup_{x\in A}$.
(with a bit of work one can recast the definition in terms of a filters and verify the axioms of a convergence space).
The "convergence almost everywhere"-structure tells us, that this convergence structure need not be topological: Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Define $P$ to be the set of measurable functions $\Omega \to [-\infty\,..\infty]$ modulo equality $\mu$-a.e. and let $f\leq g$, if $f$ is smaller then $g$ $\mu$-a.e. Then $(P,\leq)$ is a complete lattice (using "essential suprema" and "essential infima") and order convergence and convergence $\mu$-a.e. coincide, but the latter is not topological.
