Definition · applies to sets · also called $\operatorname{ext} A$

Elementary

Think of a shape drawn on paper, with an inside, an edge, and an outside. The exterior of the shape is everything outside it, not including its edge.

Undergraduate (subsets of $\reals^n$)

Consider a set $A \subseteq \reals^n$ and a point $x$. The point $x$ is in the exterior of $A$ if there exists an open ball around $x$ that does not contain any points in $A$.

Graduate (topological spaces)

Consider a subset $A$ of a topological space $X$. The exterior of $A$ is union of all open sets $E \subset X$ such that $A$ and $E$ are disjoint.

Examples

Worked examples

  • The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $\operatorname{ext} D = \setdef{(x, y) \suchthat x^2 + y^2 > 1}$ High school
    The boundary is the unit circle, which $D$ contains, and the exterior is everything outside the circle.
  • $[0, 1]$ : $\operatorname{ext} [0, 1] = (-\infty, 0) \cup (1, \infty)$ High school
    The exterior of $[0, 1]$ is everything outside $[0, 1]$, because each of those points has a small interval around it that misses $[0, 1]$.
  • $[0, 1)$, as a subset of $\reals$ : $\operatorname{ext} [0, 1) = (-\infty, 0) \cup (1, \infty)$ High school
  • $(0, 1)$ : $\operatorname{ext} (0, 1) = (-\infty, 0) \cup (1, \infty)$ High school
    The exterior of $(0, 1)$ is everything outside $[0, 1]$, because each of those points has a small interval around it that misses $(0, 1)$.
  • $\mathbb{Q}$, as a subset of $\reals$ : $\operatorname{ext} \mathbb{Q} = \emptyset$ Undergraduate

Properties

  • The exterior of a set is open.
  • The interior, boundary, and exterior of a set are disjoint, and together they make up the whole space.
  • The exterior of $A$ is the interior of its complement, $\operatorname{int}(X \setminus A)$.
  • The exterior of $A$ is not the complement of its interior.

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