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.