Show that \(\mathbb{R}\) is locally 1-euclidean and satisfies (ii) but not (i) of Exercise 2.
Solution
The usual line \(\mathbb R\) is locally \(1\)-euclidean since every point has an open interval neighborhood homeomorphic to an open interval of \(\mathbb R\). It is Hausdorff and has the countable basis of intervals with rational endpoints, so it is a \(1\)-manifold; thus it satisfies condition (ii) of E-OYPV7.
It is not compact: the open cover \[ \{(-n,n):n\in\mathbb Z_+\} \] has no finite subcover. Hence condition (i) fails. Therefore \(\mathbb R\) is locally \(1\)-euclidean and satisfies (ii) but not (i).