Thread: Topology of the Reals

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)*

Originally Posted by Chief65

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)*

(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.