Long line

This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

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.

Topological space properties

Properties it does satisfy

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