An example of a topological space. Let A be a set totally ordered by the first uncountable ordinal number. Define X = A x [0,1) \ {(a,0)}, where a is the first element of A, and order X by lexicographical order. This induces a topology on X, called the long line.

Properties of the long line:

• Locally homeomorphic to R (every point has a neighborhood homeomorphic to R). Thus, it has all the local properties of R.
• Metrizable.
• Same cardinality as R
• Has no countable basis for its topology (the points (b,0.5) have disjoint neighborhoods for all b).
• Therefore, NOT homeomorphic to R!