Boundary of a Subset of a Topological Space

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

$$\partial_X A=\operatorname{cl}_X(A)\setminus\operatorname{int}_X(A)$$

of $X$, where $\operatorname{cl}_X(A)$ is the closure of $A$ in $X$ and $\operatorname{int}_X(A)$ is the interior of $A$ in $X$. Its elements are called the boundary points of $A$ in $X$. When the ambient space $X$ is clear from context, the boundary of $A$ in $X$ is also written $\partial A$.