Interior (Topology)

Definition

Let $(X,d)$ be a metric space and $A\subseteq X$. A point $x\in A$ is an interior point of $A$ if there exists $\epsilon >0$ such that $B(x,\epsilon )\subseteq A$. The interior of $A$ is denoted $A^{\circ }$ or $\text{int}(A)$.

Properties

1. $A^{\circ }$ is the largest open set contained in $A$.
2. $A$ is open iff $A=A^{\circ }$.
3. $x\in A^{\circ }$ iff $x\notin \overline{X-A}$.