Notions of convergence not corresponding to topologies This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric? 
Answer: No.
Better Answer: Convergence ae does not even precisely correspond to a topology!
The later answer follows from the following (textbook) theorem:
Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space $X$. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.
In particular:
Corollary: If one places a topology $T$ on the measurable functions such that all the almost-everywhere convergence sequences converge in $T$, then all the convergent-in-measure sequences also converge in $T$.
Obvious questions are:


*

*Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

*Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

*Can one construct a nice theory of "convergences" different from the theory of topologies?  (Warning: This problem tortured me for some weeks some years ago.)
 A: The following two books base Analysis on convergence structures (In German only).
It was realized early in the development of Calculus beyond Banach spaces that topology is not sufficient to express the usual approach to differentiability; 
one ran into difficulties with the chain rule. 
These books are an attempt to to remedy this. 
(Many thanks to Andrew Stacey for his rubber stamp.)
Zbl 0393.46001 Gähler, Werner
Grundstrukturen der Analysis. II. (German)
Mathematische Reihe, Band 61. Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Lizenzausg. d. Akademie-Verlages, Berlin (DDR). Basel, Stuttgart: Birkhäuser Verlag. VIII, 623 S. (1978).
MSC 2010: 46-02 22A30 46A99 46G05 54A20 18A40 46A08 46M15 18D10
Zbl 0351.54001 Gähler, Werner
Grundstrukturen der Analysis. I. (German)
Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften: Mathematische Reihe. Band 58. Basel - Stuttgart: Birkhäuser Verlag. VIII, 412 S. DM 64.00 (1977).
MSC 2010: 54-01 06B05 54D05 54A05 54Bxx 54D30 18Axx 54C05 54E15 03-01 54D10
A: There is a literature on "convergence spaces" of various kinds.  I read some of that in the 70s, but I do not remember a lot of the detail.  There is something called "pre-topological space" or "closure space".   And there is "pseudo-topological space".  Each of them can be defined in terms of convergence of filters.  Or in terms of convergence of nets.  Or in terms of neighborhood systems.  Or in terms of a closure operation..  One of these is associated with Choquet.  There is a big text Topological Spaces by Cech that takes the closure space as the fundamental notion.
Added November 16.
From my old paper "Three Crypotisomorphism Theorems" in Studies in Foundations and Combinarorics, Advances in Mathematics Studies vol. 1, 1978, pp. 49--60
[Pretopology, closure space, mehrstufige Topologie, pré-adhérence]
Axioms in terms of a closure operation $\eta$ from sets to sets:
(b1) $A \subseteq \eta(A)$
(b2) $\eta(\emptyset) = \emptyset$ and $\eta(A \cup B) = \eta(A) \cup \eta(B)$.
Of course to specify a topology, we have to add a third axiom $\eta(\eta(A)) = \eta(A)$.  Without the third axiom, we get the more general pretopology.
A: Regarding your point 3 here is a list of some sources you might find interesting:


*

*Dolecki & Mynard, "Convergence Foundations of Topology": topology from convergence theory point of view, basically this is what you are asking about

*Schechter, "Hadbook of Analysis and its Foundations": good book on analysis that uses nets and filters in an essential way

*Nel, "Continuity Theory"

*Dolecki, "An invitation into Convergence Theory"


You can find more of them if you'll look for "convergence spaces" as this is a name of the object of the theory you asked for. Also you can look for "filters" and "nets".
As a side remark I can mention that there's a course on basic analysis that introduces limits using filter convergence (kind of): Zorich "Mathematical Analysis".
Also note, that the category of convergence spaces is quite general and there are some useful subcategories of it that are still more general than the topological spaces: pretopological and pseudotopological spaces.
A: As one further example of this phenomenon, allow me to recall the (apparently forgotten) category of compactologies, which was introduced and studied by Buchwalter in the 60's.  Roughly speaking, they are unions of compact spaces (with the appropriate compatibility conditions).  Every topological space has a natural compactology and every compactological space has a natural topology, but the two categories are distinct.  (They do coincide for nice spaces---metric spaces, locally compact spaces, ...).  It is clear how to define convergence of sequences for compactological spaces.  This category has two useful properties.  Firstly, for many natural spaces, the canonical topology collapses them to a point but this doesn't happen with the compactological structure.  Perhaps more importantly, this category has an intrinsic completeness concept (topological completeness is not intrinsic).  Since, as a general rule of thumb, dual objects (at least, those of interest to analysts) are always complete, there will tend to be problems with duality theory if one uses topological spaces as a basis (we are thinking, in particular, of topics like the Gelfand-Naimark duality).  In particular, we can never get a symmetric duality theory for all topological spaces (or rather completely regular spaces, since functional analysists, being interested in duality between underlying spaces and spaces of functions theoreon, tend to restrict attention to this case).  In the context of compactological spaces, these problems are less acute.
A: I answer to the point (1). In all examples one addresses the question of 

Convergence of a function (equivalently a family or a sequence) towards a certain point when the argument (equivalently the index of the family or the sequence) tends to some point. 

In my experience, there are two ways of building a topologyless theory for the argument (my answer does not cover the value side, although all sets with a filter could be endowed with a topology: just consider the discrete topology on the set, add a point $\omega$ at infinity and consider the elements of the filter as its - punctured - neighbourhoods, see e.g. Bourbaki General Topology 1). 


*

*Filters

*Nets


Filters have been invoked in a previous answer, nets are just families $(x_i)_{i\in I}$ indexed with a (filtered or directed) preordered set $(I,<)$ (i.e. two elements have an upper bound). This point of view is strictly equivalent to that of the filters (easy, but too long, exercise), however, in practice the point of view of nets is more adapted to questions with algebraic manipulations (e.g. have a look here) whereas the one of filters is more convenient for questions where domains are involved (jets, asymptotic scales, algebras of germs as there etc.).     
A: This is a well established concept in General Topology: «convergence structures». The two references I would recommend are the first chapters of each one of the following books:
E. Binz, Continuous Convergence on $C(X)$. LNM Springer, 469.
R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis.
A quick overview: On a set $X$, for every $x\in X$ we define which filters converge to $x$, with the following restrictions: the ultrafilter of all supersets of $x$ must converge to $x$; any filter which contains a filter converging to $x$ must converge to $x$; the intersection of two filters converging to $x$ must converge to $x$ («contains» and «intersection» to be understood in the usual set-theoretic sense). 
Converging filters in a convergence subspace: a filter ${\mathcal F}$ converges to $x$ in the subspace iff the filter on the initial space generated by the filter basis ${\mathcal F}$ converges to $x$.
The sets which are present in every filter converging to $x$ are known as neighborhoods of $x$ with respect to the corresponding convergence structure $\Lambda$. Such sets are actually neighborhoods in the topological sense, for a topology on $X$ (called the topology associated with $\Lambda$). The definitions of a closed set as a set whose complementary is open, and as a set which coincides with its (filterwise) adherence, are equivalent. A set $K$ is said to be $\Lambda$-compact if every ultrafilter in $K$ converges with respect to the induced convergence structure on $K$. Filterwise continuous maps send compact sets onto compact sets.
A: There is a monotone convergence property for the Cantor space that does not use topological notions here:
http://math.arizona.edu/~faris/realweb/real.pdf
See section 5 of chapter 9.
A: One can replace the words almost everywhere by on some residual set (see J. Oxtoby,  Measure and category, GTM2, Springer-Verlag). This yields another notion of convergence which is not topological.
A: Since that question popped up on the screen again I add something to the list (although I am pretty late to the party): The fact that pointwise almost everywhere convergence is not topological is nicely described in


*

*Edward T. Ordman. Convergence Almost Everywhere is Not Topological. The American Mathematical Monthly, 73(2):182–183, 1966.

A: In a locally convex topological vector space, say $E$, there are a few other notions of convergence besides the standard one which are sometimes of use.
Mackey-convergence: there is a bounded, absolutely convex set $B$ such that $(x_\gamma) \to 0$ in the normed space $E_B$ (i.e. $\bigcup \lambda B$ with the norm having $B$ as unit ball)
fast convergence: $(x_n)$ converges fast to $x$ if for each $k \in \mathbb{N}$, $n^k(x_n - x)$ is bounded.
These turn out to be quite useful in studying smooth curves and differentiation in locally convex topological spaces.  However, there won't be a topology with the property that either of these families is the family of convergent sequences.
For more on these, see A Convenient Setting of Global Analysis by Kriegl and Michor.
(I'm going to get a digital rubber stamp for the previous sentence to save me typing it every time.)
A: One example of a generalization:
A sequence in a topological space converges to a point iff for every open set around the point, cofinitely many elements of the sequence are in that open set.  Let's fixate on the words "cofinitely many" in the previous sentence.  The collection of cofinite sets is an ideal in a certain rig (ring without negation), namely the rig whose elements are subsets of the natural numbers, with multiplication given by union and addition given by intersection.  One can replace this ideal with some other ideal and get a different notion of convergence.  For instance, there is an ideal of all sets containing the number 4; using this ideal, sequence converge to their fourth elements.  Alternately, you could extend the cofinite ideal, e.g. using axiom of choice find a maximal ideal containing it.  If you use a maximal ideal, then every sequence in a compact space converges.
A: Maybe the most common form of non-topological convergence are Cesaro summability of series, and (for power series) Borel summability.  The question was about sequences, not series.  However, for sequences in a vector space, series and sequences are equivalent.
All of the examples in this thread are on the theme of sequences in a topological vector space.  The set of such sequences is a new vector space, and the examples so far are, at the very least, shift-invariant linear extensions of the map $\lim$ from convergent sequences to their limits.  Maybe any shift-invariant linear extension of $\lim$ could be called a theory of convergence?  Let's say also that the extension should be continuous in some fairly fine topology on the sequences, maybe the box topology.  If it is shift-invariant (with right shifts too if you pad with zeroes), then it automatically has the filter property that changing finitely many values does not change convergence.
Possibly linearity is not essential.  You could look at shift-invariant extensions of convergence in a topological space which are again continuous in the box topology.  Technically it should be a pointed space to define right shift, but the position of the point doesn't matter.
A: There is a notion of subsequential space:

A subsequential space is a set $X$ equipped with a relation between sequences and points, called “converges to,” with the following properties.

*

*For every $x\in X$, the constant sequence $(x)$ converges to $x$.


*If a sequence $(x_n)$ converges to $x$, then so does any subsequence of $x$.


*If, for some sequence $(x_n)$ and some point $x$, every subsequence of $(x_n)$ contains a further subsequence converging to $x$, then $(x_n)$ itself converges to $x$.

One might find a subsequential space structure on the set of measurable functions...
Note that the category of subsequential spaces is cartesian closed (which the category of all topological spaces is not), and is a full subcategory of Johnstone's topological topos (see also this blog post).
