We show that the three following axioms are equivalent in set-theory without choice, ZF:
LN : Every linearly ordered set is normal, in the order-topology.
LMN : Every linearly ordered set is monotonically normal, in the order-topology.
TOR : For every linearly ordered set, there exists an ordinal o, and a strictly increasing mapping j from X to the lexicographic order {0,1}^o.

The equivalence between LN and LMN solves a question raised by van Douwen [D85].