(a) is simply a definiton of a boundary of S in R.
(b) An intersection of closed sets is a closed set. Both (cl S) and cl (R\S) are closed sets, so bd(S) is a closed set.
I'm stuck on some proofs, wondering if anyone could lend a hand.
Let S be a subset of R
a) Prove: bd (S) = (cl S) ∩ [cl (R\S)]
b) Prove: bd (S) is a closed set
*bd (S) is a boundary point of S. cl (S) is a closure of S. (R\S) is S^c (compliment of S)*