i'm working with A(S), which is the intersection of all σ-algebras on Ω that contain S. let (S,d) be a metric space, and Ω=S, and A the collection of all open sets in (S,d).
i need to compute A(S) and show that it is the smallest possible intersection in these cases:
1) S a finite set, d the discrete metric
2) S any set, d the discrete metric
3)S=R, d the usual metric (this is the Borel sigma-algebra B(R))
4)S=R^2, d the "usual" (Euclidean distance) metric
would A(S) just be the intersection of all open sets in these respective metrics? the computation part isn't that bad, but how would one show that its the smallest possible intersection?