Complement of Set of Rational Pairs in Real Euclidean Plane is Arc-Connected

Theorem
Let $\left({\R^2, d}\right)$ be the Euclidean space of $2$ dimensions.

Let $S \subseteq \R^2$ be the subset of $\R^2$ defined as:
 * $\forall x, y \in \R^2: \left({x, y}\right) \in S \iff x, y \in \Q$

Hence let $A := \R^2 \setminus S$:
 * $\left({x, y}\right) \in A$ either $x$ or $y$ or both is irrational.

Then $A$ is arc-connected.

Proof
Let $\left({a, b}\right) \in A$.

Consider any point $\left({x_1, y_1}\right) \in A$ whose coordinates are both irrational.

By definition, either $a$ or $b$ is irrational.

WLOG suppose $a$ is irrational.

Then the union of the straight lines $x = a, y = y_1$ is an arc-connected subset of $A$ connecting $\left({x_1, y_1}\right)$ to $\left({a, b}\right)$.

Hence any point in $A$ can be connected to $\left({x_1, y_1}\right)$ by an arc.

Hence the result, by definition of arc-connected.