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!