Real Analysis/Open and Closed Sets

Terminology

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

Definition

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

We say a set $B\subset X$ is closed if $X\backslash B$ 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 $p\subset X\backslash B$, there exists a radius of $p$ which contains no point of $B$, so p cannot be a limit point of $B$, and all limit points of $B$ are contained within $B$.