Open set

Let $(X,d)$ be a metric space and let $A\subset X$.

The set $A$ is open if for every $a\in A$ there exists $\delta>0$ such that

where $B(a;\delta)$ is the open ball in $X$.

Open sets are stable under arbitrary unions and finite intersections (see basic properties of open sets ). Complements of open sets are closed .

Examples:

• In $\mathbb{R}$, every open interval $(a,b)$ is open.
• In any metric space, $\emptyset$ and $X$ are open.
• In a discrete metric space, every subset of $X$ is open.