I am reading the book "Z. Semadeni, Banach Spaces of Continuous Functions". At Definition 3.6.11, he defines a generic notion of convergence as follows:
By "upward filtering ordered sets" he means a "directed set", so his notion of net is the usual one.
Imediatelly I noticed that some requirements are missing in order to obtain some basic properties on convergent nets in topological spaces, such as:
(a) If a net is convergent then any of its subnet is convergent and they have the same set of limit points;
(b) If two convergent nets coincide in a cofinal subset of its index set, then they have at least one limit point in common;
among others. So, surely, I though: not every convergence of this kind is given by a topology. However, his comment on the almost everywhere convergence of the Lebesgue measure (highlighted in red) intrigued me, since this kind of convergence satisfies properties (a), (b) and some other properties of nets in vector topological spaces. So, here are my questions:
(1) Why is the almost everywhere convergence on the Lebesgue measure not given by any topology?
(2) Are there some sufficient conditions which guarantees that a kind of convergence is given by a topology?
Kelley General Topology (p. 74, Thm 9) mentions the following 4 conditions on net convergence to define a "convergence class" (on a set $X$):
a. If $S$ is a net such that $S_n = s \in X$ for each $n$ ($n \in N$, some directed set), then $S$ converges to $s$.
b. If $S$ converges to $s \in X$, then so does every subnet of $S$ (subnet in Kelley's sense, of course, defined on p. 70 of the book).
c. If $S$ does not converge to $s \in X$, then there is a subnet of $S$ no subnet of which converges to $s$. (A condition that a.e. convergence fails to fulfill, as shown by this argument by Ordman).
d. (what he refers to as theorem 2.4 on iterated limits): Let $D$ be a directed set, let $E_m$ be a directed set for each $m \in D$, let $F= D \times \prod_{m \in D} E_m$ (in the product (i.e. coordinatewise) ordering, which is directed too) and for each $(m,f) \in F$ let $R(m,f)=(m,f(m))$. If $\lim_m \lim_n S(m,n)=s$ ($S$ is a function into $X$ defined on all pairs $(m,n)$ with $m \in D \land n \in E_m$) for some $s \in X$ (according to the "convergence rule"), then $S \circ R$ converges to $s$ too.
The last condition is a technical one to ensure that the set of limits of nets from a set $A$, will be a valid idempotent ($\overline{\overline{A}} = \overline{A}$) closure operator. Read the book for details. All conditions a-d are valid in a topological space convergence, and if a convergence class obeys them we can define a topology for which this convergence is the topology-defined convergence.