Complex Plane is Separable

Theorem

Let $\struct {\C, \tau_d}$ be the complex number line with the usual (Euclidean) topology.

Then $\struct {\C, \tau_d}$ is separable.

Proof

Recall the definition of Separable Space:

A topological space $T = \struct {S, \tau}$ is separable if and only if there exists a countable subset of $S$ which is everywhere dense in $T$.

Consider the set:

$\Q + i \Q = \set {\alpha + i \beta : \alpha, \beta \in \Q}$

which is trivially a subset of $\C$.

We first show that $\Q + i \Q$ is countable.

Define $f : \Q^2 \to \Q + i \Q$ by:

$\map f {x, y} = x + i y$ for each $x, y \in \Q$.

This function is evidently surjective.

Further, if $x + i y = a + i b$ for $a, b \in \Q$, we have $x = a$ and $y = b$.

So $f$ is an injection.

From Rational Numbers are Countably Infinite, $\Q$ is countably infinite.

From Cartesian Product of Countable Sets is Countable, $\Q^2$ is countably infinite.

Hence there exists a bijection $g : \N \to \Q^2$.

Then $g \circ f : \N \to \Q + i \Q$ is a bijection.

Hence $\Q + i \Q$ is countably infinite.

We now show that:

$\Q + i \Q = \set {\alpha + i \beta : \alpha, \beta \in \Q}$

is everywhere dense in $\C$.

Let $x + i y \in \C$.

Let $\epsilon > 0$.

From Rationals are Everywhere Dense in Reals, there exists $\alpha \in \Q$ such that:

$\cmod {\alpha - x} < \dfrac \epsilon 2$

and:

$\cmod {\beta - y} < \dfrac \epsilon 2$

We therefore have:

Since $x + i y \in \C$ was arbitrary and $\alpha + i \beta \in \Q + i \Q$, we have that $\Q + i \Q$ is everywhere dense in $\C$.

$\blacksquare$