Show that the long line (E-PQHZN) is locally 1-euclidean and satisfies (iv) but not (iii) of Exercise 2.
Solution
Let \(L\) be the long line of E-PQHZN. By E-PQHZN, every point of \(L\) has a neighborhood homeomorphic to an open interval in \(\mathbb R\), so \(L\) is locally \(1\)-euclidean.
As a linearly ordered space in its order topology, \(L\) is a linear continuum; every linear continuum is normal. Hence \(L\) satisfies condition (iv) of E-OYPV7.
It is not metrizable. If it were metrizable, then because it is connected and locally compact, E-LI4O5 would give a countable basis on its unique component. But by E-PQHZN the long line has no countable basis (indeed no embedding into any \(\mathbb R^n\)). Therefore condition (iii) fails. Thus \(L\) satisfies (iv) but not (iii).