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.

A closed set can also be defined as a set which contains all of its limit points. This property follows directly from the previous definition, as if ${\displaystyle p\subset X\backslash B}$, there exists a radius of ${\displaystyle p}$ which contains no point of ${\displaystyle B}$, so p cannot be a limit point of ${\displaystyle B}$, and all limit points of ${\displaystyle B}$ are contained within ${\displaystyle B}$.