Interior

Definition

Suppose $X$ is a topological space and $A$ is a subset of $X$. The interior of $A$ in $X$ is defined in the following equivalent ways:

1. It is the unique largest open subset of $X$ that is contained in $A$.
2. It is the union of all open subsets of $X$ contained in $A$.
3. It is the set of all points $a\in A$ for which there exists an open subset $U\ni a$ that is completely contained in $A$.
4. It is the set-theoretic complement in $X$ of the closure of the complement of $A$ in $X$.

Related notions

• Open subset: A subset is open iff it equals its own interior.
• Regular open subset: A subset is regular open iff it equals the interior of its closure.