Long line
This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces
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 denote the minimal uncountable well-ordered set. Then , 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
- Normal space
Thus the long line fails to satisfy only the second condition for a manifold; it is simply too long.