# 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

## Contents

## 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.