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

Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

Part $\text {II}$: Counterexamples

Section $33$: Special Subsets of the Plane

Item $2$

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 $\tuple {x_1, y_1}$ with two irrational coordinates may be joined by an arc to any point $\tuple {a, b} \in A$ 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 $\tuple {x_1, y_1}$ to $\tuple {a, b}$. ...

Correction

There is a minor typo here: that last sentence should say:

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

Also see

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $33$. Special Subsets of the Plane: $2$