EDB — 0SM

[0SM] Prerequisites:[0Q0]↺↻.Let \(B(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|{\lt} r\} \) be the ball; let \(D(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|≤ r\} \) the disc; let \(S(x,r){\stackrel{.}{=}}\{ y∈ ℝ^ n : |x-y|= r\} \) be the sphere. Show that \(\overline{B(x,r)}= D(x,r)\), that \(B(x,r)= {{D(x,r)}^\circ }\), and that \(∂{B(x,r)}= S(x,r)\). Also show that \(B(x,r)\) is not closed and \(D(x,r)\) is not open.

(This result holds more generally in any normed space: see [106]↺↻).