A set \(A \subset \mathbb{R}^{n}\) is called bounded if it can be contained in a ball \(B_{r}(0) \supset A\) of finite radius \(r\). A set \(A \subset \mathbb{R}^{n}\) is called pathwise connected if we can go along a curve from any point \(a \in A\) to any other point \(a^{\prime} \in A\) without ever leaving \(A\).

(i) Construct an open and unbounded set in \(\mathbb{R}\) with finite, strictly positive Lebesgue measure.

[ try unions of ever smaller open intervals centred around \(n \in \mathbb{N}\).]

(ii) Construct an open, unbounded and pathwise connected set in \(\mathbb{R}^{2}\) with finite, strictly positive Lebesgue measure.

[ try a union of adjacent, ever longer, ever thinner rectangles.]