Definition 9

In a topological space $X$, a point $x$ is said to be an interior point of a subset $A$ of $X$ if $A$ is a neighbourhood of $x$. The set of interior points of $A$ is called the interior of $A$ and is denoted by $\dot{A}$.

According to Definitions 9 and 4, a point $x$ is an interior point of $A$ if there is an open set contained in $A$ which contains $x$; it follows that $\dot{A}$ is the union of all the open sets contained in $A$, and hence is the largest open set contained in $A$: in other words, if $B$ is an open set contained in $A$, then $B \subset \dot{A}$. Consequently, if $A$ and $B$ are two subsets of $X$ such that $B \subset A$, then $\dot{B} \subset \dot{A}$; and $A$ is a neighbourhood of $B$ if and only if $B \subset \dot{A}$.