It is known that the Axiom of Choice implies the following axiom:
LN: Every linearly ordered set is normal in the order-topology.
In this paper, we prove that in Cohen's first model, every linearly ordered set is normal. Hence the axiom LN does not imply the axiom of choice.

This solves a question raised by Birkhoff (1940), which was left open in van Douwen's paper, [D85] and also staid open in my thesis, [Mo88b].