[:compact in R^n iff closed + bounded:] • JRM:Top.2 §27.3 • Take some subspace topology ARnA \subseteq \mathbb R^n. Then TFAE: • AA is compactAA is closed and AA is bounded using the euclidean metric ddAA is closed and AA is bounded using the square metric ρ\rho • Equivalence of the final two essentially come from that ρdρn\rho \leq d \leq \rho \sqrt n; "the two metrics dominate each other" and so boundedness is biconditional Referenced by: