# Real Analysis/Open and Closed Sets

## Terminology

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

## Definition

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.