Long line

Definition
The long line is defined as follows: Let $$S_\Omega$$ denote the minimal uncountable well-ordered set. Then $$L = S_\Omega \times [0,1)$$, in the dictionary order, is the long line.

Properties it does not satisfy

 * Second-countable space: The long line is not second-countable
 * Sub-Euclidean space: The long line cannot be embedded inside any Euclidean space

Properties it does satisfy

 * Linearly orderable space: It is defined using a linear order, so it is clearly linearly orderable
 * Locally Euclidean space: It is in fact locally homeomorphic to $$\R^1$$
 * Normal space

Thus the long line fails to satisfy only the second condition for a manifold; it is simply too long.