# Interior **The largest open [[subset]] of a topological space that is completely contained by a subset.** ![[Interior.svg]] > [!NOTE] Interior > The *interior* of a subset $S$ of a topological space $X$ can be equivalently defined as: > - the largest open subset of $X$ that is completely contained in $S$. > - the [[union]] of all open subsets of $X$ completely contained in $S$. > - the set of all interior points of $S$. > [!NOTE] Interior point > If $S$ is a subset of a topological space $X$, then a point $p$ in $X$ is an *interior point* of $S$ if there exists an open set centred at $p$ which is completely in $S$. > > Equivalently, $p$ is an interior point of $S$ if $S$ is a [[neighbourhood]] of $p$.