Closed Ball is Closed

Last updated Nov 1, 2022

Let $(M, d)$ be a Metric Space. Let $x \in M$ and $\epsilon > 0$. Then $\overline{B_{\epsilon}(x)}$ is Closed.

Consider $\overline{B_{\epsilon}(x)}^{C} = {x’ \in M: d(x’, x) > \epsilon}$. For any $x’$ in this Set, let $\delta := d(x’, x) - \epsilon$. If $y \in B_{\delta}(x’)$, then $$d(x, y) + d(x, x’) - \epsilon = d(x, y) + \delta > d(x, y) + d(y, x’) \geq d(x, x’).$$ Therefore, $d(x, y) > \epsilon$. Thus $B_\delta(x’) \subset \overline{B_{\epsilon}(x)}^{C}$. Because A Set is Open in the Metric Topology iff it contains a Ball around each Point, $\overline{B_{\epsilon}(x)}^{C}$ is Open. Therefore $\overline{B_{\epsilon}(x)}$ is Closed. $\blacksquare$

• Closed Ball
• Open Ball