When is an ordered field a metric space?

Abstract

Let $(F, \leq)$ be an ordered field. With respect to the order topology, $F$ is a Tychonoff uniform space. $F$ is metrizable if and only if there is a countable set $\{b_{1}, \ldots, b_{n}, \ldots\}$ of positive elements of $F$ such that if $b$ is any positive element of $F$, there exists $n\geq 1$ such that $0 \lt b_{n} \lt b$. If $F$ is denumerable or Archimedean, then this metrizability condition is satisfied. For each uncountable cardinal number $\aleph$, there exist ordered fields, $F_{1}$ and $F_{2}$, each of cardinality $\aleph$, such that the order topology on $F_{1}$ (resp., $F_{2}$) is (resp., is not) metrizable.