Interior

The set of all points $x$ of a subset $A$ of a topological space $X$ for which an open set $U_x$ in $X$ exists such that $x \in U_x$ and $U_x \subseteq A$. The interior of the set $A$ is usually denoted by $\mathrm{Int}\, A $ and represents the largest open set in $X$ contained in $A$. The equality $\mathrm{Int}\, A = X \setminus [X \setminus A]$ holds, where $[]$ denotes closure in $X$. The interior is sometimes called the open kernel of the set.

The interior of a closed set in a topological space $X$ is a regular open or canonical set. Spaces in which the open canonical sets form a base for the topology are called semi-regular. Every regular space is semi-regular.

Comments

See also Interior of a set.