Real Analysis/Open and Closed Sets

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

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

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