# Complement, closure, interior.
Let $(X,d)$ be a metric space, and let $A \subset X$ be any subset.
A point $p \in X$ is called an **interior point** of $A$ if there exists an open ball $B(p,r)$ such that $B(p,r) \subset A$.
A point $p \in X$ is called an **adherent point** of $A$ if every open ball $B(x,r)$ intersects $A$.
We say the **interior** of $A$ to be the set of all interior points of $A$, denote as $\text{int}(A)=A^{o}$.
We say the **closure** of $A$ to be the set of all adherent points of $A$, denote as $\text{cl}(A) = A^{k}$.
Let us denote the **complement** of $A$ to be $\text{co}(A)=A^{c}=\{x\in X : x \notin A \}$.