Open set

Let $(X,d)$ be a metric space . A subset $U\subseteq X$ is open if for every $x\in U$ there exists $r>0$ such that

Open sets are the primitive “admissible neighborhoods ” in topology. In analysis, openness is the natural condition for local arguments (e.g., differentiability is typically defined on open subsets of $\mathbb{R}^k$).

Examples:

• In $\mathbb{R}$, every open interval $(a,b)$ is open.
• In $\mathbb{R}^k$, every open ball $B(x,r)$ is open.
• In any metric space, $\varnothing$ and $X$ are open.