Basic Analysis I — Real Analysis · Problem 7.2.18

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · open-and-closed-sets

Problem

For every x∈Rnx \in \mathbb{R}^n and every δ>0\delta > 0 define the ``rectangle'' R(x,δ):−(x1−δ,x1+δ)×(x2−δ,x2+δ)×⋯×(xn−δ,xn+δ)R(x,\delta) \coloneq (x_1-\delta, x_1+\delta) \times (x_2-\delta, x_2+\delta) \times \cdots \times (x_n-\delta, x_n+\delta). Show that these sets generate the same open sets as the balls in standard metric. That is, show that a set U⊂RnU \subset \mathbb{R}^n is open in the sense of the standard metric if and only if for every point x∈Ux \in U, there exists a δ>0\delta > 0 such that R(x,δ)⊂UR(x,\delta) \subset U.

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