Show that \(\mathbb{R} \times \mathbb{R}\) in the dictionary order topology is locally 1-euclidean and satisfies (iii) but not (ii) of Exercise 2.
Solution
Let \(X=\mathbb R\times\mathbb R\) with the dictionary order. For \((a,b)\in X\), the order interval \[ ((a,b-1),(a,b+1)) \] is exactly \(\{a\}\times(b-1,b+1)\), and projection onto the second coordinate is an order-preserving homeomorphism with \((b-1,b+1)\). Thus \(X\) is locally \(1\)-euclidean.
The dictionary-order plane is metrizable by E-KY2W7, so it satisfies condition (iii) of E-OYPV7.
It does not satisfy condition (ii). Indeed the sets \[ U_a=\{a\}\times(0,1)\qquad(a\in\mathbb R) \] form an uncountable family of pairwise disjoint nonempty open subsets. A second-countable space cannot have such a family: each member would contain a distinct element of a fixed countable basis. Hence \(X\) is not second-countable, so it is not a \(1\)-manifold. Thus it satisfies (iii) but not (ii).