Boundary and exterior organize a metric space around a subset. The interior contains points safely inside the set, the exterior contains points safely outside, and the boundary contains points approached by both the set and its complement. These concepts complete the basic topological structure of metric spaces. :::definition[Exterior] Let $(X,d)$ be a metric space and let $A\subset X$. The $exterior$ of $A$ is $\operatorname{ext}(A)=(X\setminus A)^\circ$. ::: :::definition[Boundary] Let $(X,d)$ be a metric space and let $A\subset X$. The $boundary$ of $A$ is $\partial A=\overline{A}\cap\overline{X\setminus A}$. ::: :::example[Boundary of an Interval] For $A=(0,1)\subset\mathbb{R}$, $A^\circ=(0,1)$, $\operatorname{ext}(A)=(-\infty,0)\cup(1,\infty)$, and $\partial A=\{0,1\}$. ::: :::example[Boundary of the Rationals] For $A=\mathbb{Q}\subset\mathbb{R}$, both $A$ and its complement are dense, so $\partial A=\mathbb{R}$. ::: :::theorem[Boundary Identities] Let $(X,d)$ be a metric space and let $A\subset X$. Then $\partial A=\partial(X\setminus A)$, $\overline{A}=A^\circ\cup\partial A$, $A^\circ\cap\partial A=\varnothing$, and $X=A^\circ\cup\partial A\cup(X\setminus A)^\circ$. ::: :::proof Given that $A\subset X$. To prove the boundary identities. First, $$\begin{align*} \partial(X\setminus A) &=\overline{X\setminus A}\cap\overline{A}\\ &=\partial A. \end{align*}$$ Since $A^\circ\subset\overline{A}$ and $\partial A\subset\overline{A}$, we have $A^\circ\cup\partial A\subset\overline{A}$. If $x\in\overline{A}$ and $x\notin A^\circ$, then $x\in X\setminus A^\circ=\overline{X\setminus A}$, so $x\in\partial A$. Hence $\overline{A}=A^\circ\cup\partial A$. If $x\in A^\circ$, then some ball around $x$ lies in $A$, so $x\notin\overline{X\setminus A}$ and $x\notin\partial A$. Thus $A^\circ\cap\partial A=\varnothing$. The decomposition of $X$ follows from $X\setminus\overline{A}=(X\setminus A)^\circ$. $\square$ ::: :::solved-problem Give an example where $(\overline{Y})^\circ\neq\overline{Y^\circ}$. ::: :::solution Let $Y=\mathbb{Q}\subset\mathbb{R}$. Then $\overline{Y}=\mathbb{R}$, so $(\overline{Y})^\circ=\mathbb{R}$. But $Y^\circ=\varnothing$, so $\overline{Y^\circ}=\varnothing$. Hence the two sets are not equal. $\square$ ::: :::exercise [1] State the main definition from this lesson. [2] State the main theorem from this lesson. [3] Give one example illustrating the theorem. [4] Explain one common mistake related to this topic. ::: :::answer [1] The main definition is the first titled definition in the lesson. [2] The main theorem is the titled theorem proved in the lesson. [3] The examples in the lesson provide standard choices. [4] A common mistake is to ignore the metric or the ambient space when applying the definition. ::: :::faq[Frequently Asked Questions] Q: Why is the ambient metric space important? A: Openness, closedness, closure, and boundary are interpreted relative to the chosen metric space. Q: Can the same set behave differently in different metric spaces? A: Yes. Changing the metric or the ambient space may change topological behaviour. Q: How should students check a definition? A: They should return to balls, closures, and sequences, depending on the context. ::: :::call-to-action[Continue from Boundary and Exterior of Sets] subtitle: Review the definitions and proofs before moving through the metric topology sequence. button: Section Start | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/metric-balls-and-neighbourhoods button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/oscillation-and-sets-of-discontinuity :::