ABSTRACT
Therefore y ∈ Nr(x0), and soNρ(x) ⊂ Nr(x0). Since x ∈ Nr(x0) was arbitrary, Nr(x0) is open. "
! 2.25 For any real r > 0 and any x0 ∈ X, show that N ′r(x0) is an open set. ! 2.26 Using Definition 3.1, prove that every interval of the form (a, b) ⊂ R where a < b is an open set in R. Handle as well the cases where a is replaced by −∞ or b is replaced by∞. ! 2.27 Using Definition 3.1, prove that every open interval of the form (a, b)×(c, d) ⊂ R 2 where a < b and c < d is an open set in R2. Extend this result to open intervals
(a1, b1)× (a2, b2)× · · ·× (ak, bk) in Rk. Consider the infinite intervals of Rk as well.
