If anyone helps me with these questions, I will be thankful.
(a) Show that every closed set in a metric space can be expressed as an
intersection of countably many open sets.
(b) Show that every open set in a metric space can be expressed as an
union of countably many closed sets. (Hint: take complements.)
(c) Express the interval (0, 1) in R as a union of countably many open
sets.
Thank you for your help.
Will it be OK if I answer the part (a) like this?
Let A_t be an open set in (M,d).
Then, A_t = { d (a,x) < t for some a in A }
So, ∩ { d (a,x) < 1/n for some a in A } = cl (A) which is the smallest closed set containing A.
If it is true, how can I modify it for the answer of part (b)?