Sphere is Closed

Last updated Nov 1, 2022

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

Closed Ball is Closed and $S_{\epsilon}(x) = \overline{B_{\epsilon}(x)} \setminus B_{\epsilon}(x) = \overline{B_{\epsilon}(x)} \cap B_{\epsilon}(x)^{C}$. $\blacksquare$