In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.

The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

If ${\displaystyle S}$ is a subset of a Euclidean space, then ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if there exists an open ball centered at ${\displaystyle x}$ which is completely contained in ${\displaystyle S.}$ (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset ${\displaystyle S}$ of a metric space ${\displaystyle X}$ with metric ${\displaystyle d}$: ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if there exists a real number ${\displaystyle r>0,}$ such that ${\displaystyle y}$ is in ${\displaystyle S}$ whenever the distance ${\displaystyle d(x,y)<r.}$

This definition generalizes to topological spaces by replacing "open ball" with "open set". If ${\displaystyle S}$ is a subset of a topological space ${\displaystyle X}$ then ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ in ${\displaystyle X}$ if ${\displaystyle x}$ is contained in an open subset of ${\displaystyle X}$ that is completely contained in ${\displaystyle S.}$ (Equivalently, ${\displaystyle x}$ is an interior point of ${\displaystyle S}$ if ${\displaystyle S}$ is a neighbourhood of ${\displaystyle x.}$)

The interior of a subset ${\displaystyle S}$ of a topological space ${\displaystyle X,}$ denoted by ${\displaystyle \operatorname {int} _{X}S}$ or ${\displaystyle \operatorname {int} S}$ or ${\displaystyle S^{\circ },}$ can be defined in any of the following equivalent ways:

If the space ${\displaystyle X}$ is understood from context then the shorter notation ${\displaystyle \operatorname {int} S}$ is usually preferred to ${\displaystyle \operatorname {int} _{X}S.}$

On the set of real numbers, one can put other topologies rather than the standard one: