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Topology of the Complex Plane#

Theorem: Topology of the Complex Plane

The function \(d: \mathbb{C} \times \mathbb{C} \to \mathbb{R}\) defined as

\[ d(z_1, z_2) = |z_2 - z_1| \]

is a metric on the complex numbers.

Proof

TODO

Note

As a metric, \(d\) induces a topology on \(\mathbb{C}\) and, unless otherwise specified, all topology-related terminology relates to this induced topology.

Definition: \(r\)-Neighborhood

Let \(r \gt 0\) and \(z \in \mathbb{R}\).

The \(r\)-neighborhood of \(z\) is the open ball \(B_r(z)\), i.e.

\[ B_r(z) = \{x \in \mathbb{C} \mid |x - z| \lt r\} \]

Definition: Deleted \(r\)-Neighborhood

Let \(r \gt 0\) and \(z \in \mathbb{R}\).

The deleted \(r\)-neighborhood of \(z\) is its \(r\)-neighborhood without \(z\) itself:

\[ B_r(z) \setminus \{z\} = \{x \in \mathbb{C} \mid 0 \lt |x - z| \lt r \} \]

Theorem: Boundary and Interior of Neighborhoods

Let \(r \gt 0\) and \(z \in \mathbb{C}\).

The boundary of the \(r\)-neighborhood and the deleted \(r\)-neighborhood of \(z\) is the disk

\[ \{ x \in \mathbb{C} \mid |x - z| = r \} \]
Proof

TODO