Real Analysis/Open and Closed Sets

Terminology

The open ball in a metric space (X,d) with radius ε centered at a, is denoted $B(a, \epsilon)$. Formally $B(a, \epsilon) = \{x \in X: d(a,x) < \epsilon\}$

Definition

Let (X,d) be a metric space. We say a set $A \subset X$ is open if for every $x \in A \text{ } \exists \epsilon > 0$ such that $B(x,\epsilon) \subset A$.

We say a set $B \subset X$ is closed if $X\backslash B$ is open.