Real Analysis/Open and Closed Sets

Terminology
The open ball in a metric space $$(X,d)$$ with radius $$ \epsilon $$ 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.