 Boundary of a set

The boundary of a subset $A$ of a topological space $X$ consists of the points whose every neighbourhood in $X$ meets both $A$ and its complement $X\setminus A$. Equivalently,Here the two operations are closure and interior. The second formula shows that the boundary is a closed set and that $A$ and its complement have the same boundary. It locates precisely the points where membership cannot be decided from a small neighbourhood alone. ^[ref]

For $A=[0,1)$ in the usual real line, the interior is $(0,1)$, the closure is $[0,1]$, and the boundary is $\{0,1\}$. In particular, $0$ belongs both to $A$ and to its boundary, while $1$ belongs to the boundary but not to $A$. A boundary point need not be excluded from the original set: open sets exclude all their boundary points, closed sets contain all theirs, and this half-open interval does one of each.

The upper panel visualizes the relative example from open set: for $X=[0,1]$ and $A=[0,1/2)$, an open ball about zero in $X$ is $B_X(0,r)=X\cap (-r,r)=[0,r)$ when $0<r<1/2$. This open ball stays in $A$; points outside $X$ are not required. The middle panel shows that $X$ is clopen in itself, with interior and closure both $X$ and empty boundary, although its boundary in the real line is $\{0,1\}$. The lower panel returns to ambient space $\mathbb{R}$ and separates the interior, closure, and boundary of $[0,1)$; both marked boundary points are included in the boundary itself.