When does it make sense to define a generator of a set system?

Question

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system containing it. Note that "the coarsest" is an optimum object, which in general may or may not exist. So I am not sure if a set system always admits a generator? When does a set system admit a generator, and when doesn't? Are there some examples of a set system that doesn't admit a generator?

One way I just realize is that for a kind of set systems to admit a generator, it suffices that the intersection of arbitrarily many such set systems still a set system of the same kind. In other words, the family of all set systems on the same ground set must be closed under arbitrary intersection. Am I right? Is it a necessary condition for existence of a generator? What kinds of set systems are such that all such set systems on a grounding set are closed under arbitrary intersection, and what kinds are not?

Added: A link says:

Any type of algebraic structure on subsets of $S$ that is defined purely in terms of closure properties will be preserved under intersection. That is, we will have results that are analogous to how σ-algebras are generated from more basic sets, with completely straightforward and analgous proofs. In the following two theorems, the term system could mean π-system, λ-system, or monotone class of subsets of S.

...

Note however, that Theorems 3 and 4 do not apply to semi-algebras, because the semi-algebra is not defined purely in terms of closure properties (the condition on $A^c$ is not a closure property).

So I wonder what "closure properties" mean in the quote? What does "the condition on $A^c$ is not a closure property" mean?

Thanks and regards!

Answer 1

What are the properties of generator sets? Let $\cal B$ be a "system", if $A$ is a generator set for $\cal B$ then we expect the following two properties:

1. Whenever $\cal B'$ is a "system" with $A\subseteq\cal B'$ then $\cal B\subseteq B'$. Namely, $\cal B$ is the smallest "system" containing $A$.

2. If we close $A$ under the operations of the "system" we shall arrive at $\cal B$. So every element in $\cal B$ has a recipe of how to generate it from elements of $A$.

In the case of a $\sigma$-algebra this is quite obvious as to why this sort of system can have generator sets. We have very nice operations to close under, and the extensionality of $\in$ imply that generating sets using unions and intersections and completion (with respect to a fixed space, of course) is a uniquely determined process.

The case of a topology is not very different. In here we require closure under unions and finite intersections. So by simply closing under finite intersections and arbitrary unions, again the extensionality of $\in$ assures that this is going to be a uniquely determined system.

I can't give much insight with convexity structures, but I hope the arguments above somehow help to clear it up. If we have some sort of guarantee that the operations do not change from passing from one system to a smaller system, then generators would be easy to come up with. Much like addition does not change when passing from $\mathbb Q$ to $\mathbb Z$, this allows us to find a generator for $\mathbb Z$ as a group.

Answer 2

This works for Moore closures. You have a family $\mathcal{C}$ of subsets of a set $X$ that contains the whole set and is closed under arbitrary intersections. Such a family is a Moore collection. This defines a closure $C:2^X\to 2^X$ by $$C(G)=\bigcap_{C\in\mathcal{C},C\supseteq G}C.$$ We can then call $G$ a generator of $C(G)$. It is easily seen that $C$ is idempotent, monotone, and satisfies $G\subseteq C(G)$ for all $G\subseteq X$. One can show that every such function $C:2^X\to 2^X$ comes from some Moore collection as described above.

This framework generalizes all the constructions described so far. Related to Asafs answer, one can show that all Moore closures arise from closing a set under some (potentially infintary) operations. The details are in Chapter 4 of the Handbook of Analysis and its Foundations by Eric Schechter.