## Tsukuba Journal of Mathematics

### When is an ordered field a metric space?

David E. Dobbs

#### 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.

#### Article information

Source
Tsukuba J. Math., Volume 24, Number 2 (2000), 325-336.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164154

Digital Object Identifier
doi:10.21099/tkbjm/1496164154

Mathematical Reviews number (MathSciNet)
MR1818091

Zentralblatt MATH identifier
1065.12003

#### Citation

Dobbs, David E. When is an ordered field a metric space?. Tsukuba J. Math. 24 (2000), no. 2, 325--336. doi:10.21099/tkbjm/1496164154. https://projecteuclid.org/euclid.tkbjm/1496164154