Is the boundary of a boundary a subset of the boundary? The definition of a boundary of a set $S$ in a topological space $X$ is $\text{comp}\{\text{Int}(S) \cup \text{Ext}(S)\}$ (complement of the interior union exterior).
The definition for interior is the set of all interior points.
The definition for interior point of $S$ is if there exists an open neighborhood $N$ of that point such that $N \subset S$.
The definition of neighborhood of a point $x$ is a subset of the topological space $X$ that contains an open set such that $x$ is in that open set.
The definition for exterior of $S$ is $\text{Int}(\text{Comp}(S))$.
So $\text{Bdry}(\text{Bdry}(S)) = \text{comp}\{\text{Int}(\text{comp}\{\text{Int}(S) \cup \text{Ext}(S)\}) \cup \text{Ext}(\text{comp}\{\text{Int}(S) \cup \text{Ext}(S)\})\}$ which I can't make heads or tails of. Is there an easier approach to checking of the boundary of the boundary is a subset of the boundary using these definitions?

The boundary of $S$ is closed in $X$, because it is the intersection of two closed subsets: $$\overline{S} \cap (X \backslash \mathrm{Int}(S)).$$ Therefore, it contains its own boundary.
