Long line

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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 $S_{\mathrm{\Omega }}$ denote the minimal uncountable well-ordered set. Then $L=S_{\mathrm{\Omega }}\times [0,1)$, in the dictionary order, is the long line.

Topological space properties

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 ${\mathbb{R}}^1$
• Normal space

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