Reading about random sets, I've come across the phrase " B ( F ) is the Borel σ-algebra generated by the topology of closed convergence." where F is the family of closed sets in R .I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel σ-algebra on a set S is the σ-algebra generated by the open sets of S, 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".

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Reading about random sets, I&#039;ve come across the phrase &quot;      B    (      F    ) is the Borel   σ-algebra generated by the topology of closed convergence.&quot; where       F   is the family of closed sets in       R    .I don&#039;t know a lot about topology, so I&#039;m not sure how to understand/interpret this. In general, a Borel   σ-algebra on a set   S is the   σ-algebra generated by the open sets of   S, and I suppose a topology gives a notion of &quot;openness&quot;, but I would appreciate if someone could provide me some intuition behind this particular σ-algebra and the &quot;topology of closed convergence&quot;.

Patricia Curry · Accepted Answer

For any family       F   of subsets of a set   X we can talk about the   σ-algebra generated by       F   on   X, which is the defined to be the intersection of all   σ-algebras on   X that contain       F   as a subset. This is standard in measure theory. It&#039;s well-defined as the intersection of   σ-algebras is a   σ-algebra again and we always have at least one σ-algebra containing       F  , namely the power set of   X.A topology on   X is just another such family       F   so the definition applies. The   σ-algebra will thus contain all open sets, all closed sets (their complements), all       G    δ   sets and       F    σ   sets etc. (all in that topology). It&#039;s also called the Borel   σ-algebra of a topological space.

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