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

Source Work

 * Part $\text{II}$: Counterexamples
 * Section $33$: Special Subsets of the Plane
 * Item $2$
 * Item $2$

This mistake can be seen in the second edition (1978) as republished by Dover in 1995: ISBN 0-486-68735-X

Mistake

 * Let $A$ be the subset of $\R^2 = \R \times R$ consisting of all point with at least one irrational coordinate, and let $A$ have the induced topology. $A$ is arc-connected since a point $\left({x_1, y_1}\right)$ with two irrational coordinates may be joined by an arc to any point $\left({a, b}\right) \in A$ either $a$ or $b$ is irrational, say $a$. Then the union of the lines $x = a, y = y_1$ is an arc-connected subset of $X$ connecting $\left({x_1, y_1}\right)$ to $\left({a, b}\right)$. ...

There is a minor typo here: that last sentence should say:
 * Then the union of the lines $x = a, y = y_1$ is an arc-connected subset of $A$ connecting $\left({x_1, y_1}\right)$ to $\left({a, b}\right)$.

Also see

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