Long line

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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 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 not satisfy

Properties it does satisfy

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