Reading about random sets, I've come across the phrase " B ( F ) is...
Santino Bautista
Answered question
2022-06-21
Reading about random sets, I've come across the phrase " is the Borel -algebra generated by the topology of closed convergence." where is the family of closed sets in
I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel -algebra on a set is the -algebra generated by the open sets of , and I suppose a topology gives a notion of "openness", but I would appreciate if someone could provide me some intuition behind this particular σ-algebra and the "topology of closed convergence".
I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel -algebra on a set is the -algebra generated by the open sets of , and I suppose a topology gives a notion of "openness", but I would appreciate if someone could provide me some intuition behind this particular σ-algebra and the "topology of closed convergence".
Answer & Explanation
Patricia Curry
Beginner2022-06-22Added 15 answers
For any family of subsets of a set we can talk about the -algebra generated by on , which is the defined to be the intersection of all -algebras on that contain as a subset. This is standard in measure theory. It's well-defined as the intersection of -algebras is a -algebra again and we always have at least one σ-algebra containing , namely the power set of .
A topology on is just another such family so the definition applies. The -algebra will thus contain all open sets, all closed sets (their complements), all sets and sets etc. (all in that topology). It's also called the Borel -algebra of a topological space.
A topology on is just another such family so the definition applies. The -algebra will thus contain all open sets, all closed sets (their complements), all sets and sets etc. (all in that topology). It's also called the Borel -algebra of a topological space.
