Measure, Integration & Real Analysis · Problem 5C.1

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · open-sets-rn-euclidean-norm

Problem

Show that a set G⊆RnG \subseteq \mathbf{R}^n is open in Rn\mathbf{R}^n if and only if for each (b1,…,bn)∈G(b_1, \ldots, b_n) \in G, there exists r>0r > 0 such that {(a1,…,an)∈Rn:(a1−b1)2+⋯+(an−bn)2<r}⊆G.\left\{(a_1, \ldots, a_n) \in \mathbf{R}^n : \sqrt{(a_1 - b_1)^2 + \cdots + (a_n - b_n)^2} < r\right\} \subseteq G.

Related theory

Browse the problem library