Decomposition of a Topological Space by the Boundary of a Subset

Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$. Write $X\setminus S$ for the complement relative to $X$ of a subset $S\subseteq X$, let $\operatorname{int}_X$ and $\operatorname{cl}_X$ denote the interior and the closure in $X$, and let $\partial_X$ denote the boundary in $X$.

Then the following hold.

1. $\partial_X A=\operatorname{cl}_X(A)\cap\operatorname{cl}_X(X\setminus A)$.

2. $\partial_X A$ is closed in $(X,\mathcal{T})$.

3. $\partial_X A=\partial_X(X\setminus A)$.

4. The three sets $\operatorname{int}_X(A)$, $\partial_X A$ and $\operatorname{int}_X(X\setminus A)$ are pairwise disjoint, and their union is $X$.

5. The two sets $\operatorname{int}_X(A)$ and $\partial_X A$ are disjoint, and their union is $\operatorname{cl}_X(A)$.