If S is a subset of the Reals Prove:
a) bd(S) = bd(R\S)
b) int(S) intersect bd(S) = empty set
c) S is open IFF bd(S) is a subset of R\S
a) A point is a boundary point of iff every neighborhood around contains points in and . A point is a boundary point of iff every neighborhood around contains points in and . What can you conclude from this?
b) A point is an interior point of iff there exists a neighborhood around that is a proper subset of . Compare this definition with the above definition and deduce that .
c) If is open, then every point in is an interior point, so . is trivially true.
If , then , meaning that every point in is an interior point, so is open.