Equivalence of Definitions of Exterior Point (Complex Analysis)

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Theorem

The following definitions of the concept of exterior point in the context of Complex Analysis are equivalent:


Let $S \subseteq \C$ be a subset of the complex plane.

Let $z_0 \in \C$.

Definition 1

$z_0$ is an exterior point of $S$ if and only if $z_0$ has an $\epsilon$-neighborhood which is disjoint from $S$.

Definition 2

$z_0$ is an exterior point of $S$ if and only if:

$z_0$ is not an interior point of $S$

and:

$z_0$ is not a boundary point of $S$.


Proof

Let $S \subseteq \C$.


Definition 1 implies Definition 2

Let $z_0$ be an exterior point of $S$ by definition 1.

Let $N_\epsilon \left({z_0}\right)$ be an $\epsilon$-neighborhood of $z_0$ such that $N_\epsilon \left({z_0}\right) \cap S = \varnothing$.

That is, $N_\epsilon \left({z_0}\right)$ has no points which are also in $S$.

By definition, it follows that $z_0$ is not a boundary point of $S$.


Aiming for a contradiction, suppose $z_0$ is an interior point of $S$.

Let $N_{\epsilon'} \left({z_0}\right)$ be an $\epsilon'$-neighborhood such that $N_{\epsilon'} \left({z_0}\right) \subseteq S$.

By Empty Intersection iff Subset of Complement:

$N_\epsilon \left({z_0}\right) \subseteq \complement_\C \left({s}\right)$



Thus $z_0$ is an exterior point of $S$ by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $z_0$ be an exterior point of $S$ by definition 2.

As $z_0$ is not an interior point of $S$ there exists no $\epsilon$-neighborhood of $z_0$ which is disjoint from $\complement_\C \left({S}\right)$.

That is, every $\epsilon$-neighborhood of $z_0$ contains points which are not in $S$.

As $z_0$ is not a boundary point of $S$, there exists at least one $\epsilon$-neighborhood of $z_0$ which does not contain both points in $S$ and points not in $S$.

But as every $\epsilon$-neighborhood of $z_0$ contains points which are not in $S$, it follows there must be at least one $\epsilon$-neighborhood of $z_0$ disjoint from $S$.

Thus $z_0$ is an exterior point of $S$ by definition 1.

$\blacksquare$