EDB — 0QC

Exercises

1. [0QC] Prerequisites:[0R9]↺↻.Show that, for every closed set \(C⊆ X\) there exist countably many open sets \(A_ n\) such that \(⋂_ n A_ n=C\).

   Solution 1

   [0QD]↺↻

   A set obtained as an intersection of countably many open sets is known as ”a \(G_𝛿\) set”. The previous exercise shows that in a metric space every closed is a \(G_𝛿\).

   Passing to the complement set, one obtains this statement. A set that is union of countably many closed sets is known as ”an \(F_𝜎\) set”. The previous exercise shows that in a metric space every open set is an \(F_𝜎\) set.

   See also the section [14J]↺↻.