Interior of a Subset of a Topological Space

Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$. The interior of $A$ in $X$ is the subset

$$\operatorname{int}_X(A)=\{x\in X:\ \text{there is } U\in\mathcal{T} \text{ with } x\in U \text{ and } U\subseteq A\}$$

of $X$. Its elements are called the interior points of $A$ in $X$. When the ambient space $X$ is clear from context, the interior of $A$ in $X$ is also written $A^{\circ}$.