Real Analysis/Open and Closed Sets

Terminology

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

Definition

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

We say a set $B\subset X$ is closed if $X\mathrm{\setminus }B$ is open.