Interior of Set of Real Numbers in Complex Numbers is Empty

Theorem

Let $\tuple {\C, d}$ be the complex Euclidean space.

Let $\R$ be the subspace of real numbers.

Then the interior of $\R$ in $\C$ is the empty set $\O$.

Let $\tuple {\C, d}$ be the complex Euclidean space.

Consider $S \subseteq \R$ as a topological subspace of $\tuple {\C, d}$.

Then the interior of $S$ in $\C$ is the empty set $\O$.

Aiming for a contradiction, suppose that:

$\R^\circ \ne \O$

Let $x \in \R^\circ$.

From the definition of an open subset of $\C$, there exists $\epsilon > 0$ such that:

$\set {z \in \C : \cmod {z - x} < \epsilon} \subseteq \R^\circ$

Consider:

$\ds z = x + \frac \epsilon 2 i \in \C \setminus \R$

Then, we have:

$\ds \cmod {z - x} = \cmod {x + \frac \epsilon 2 i - x} = \frac \epsilon 2 < \epsilon$

Hence $z \in \R^\circ$.

But by the definition of interior, we have $\R^\circ \subseteq \R$.

This contradicts the fact that $z \in \C \setminus \R$.

$\blacksquare$