alternative definition of metric space
1 Definition
The notion of metric space may be defined in a way which makes use only of rational numbers^{} as opposed to real numbers. To avoid mention of real numbers we take a three place relation^{} as our primitive term instead of the distance function. Thus, one can use this definition to define the set of real numbers as the completion of the set of rational numbers as a metric space. Let ${\mathbb{Q}}_{+}$ denote the set of strictly positive rational numbers
Definition 1.
A metric space consists of a set $X$ and a relation $D\mathrm{\subset}X\mathrm{\times}X\mathrm{\times}{\mathrm{Q}}_{\mathrm{+}}$ which satisfies the following properties:

1.
For all $x,y\in X$, it is the case that $D(x,y,r)$ for all $r\in {\mathbb{Q}}_{+}$ if and only if $x=y$.

2.
For all $x,y\in X$ and all $r\in {\mathbb{Q}}_{+}$, it is the case that $D(x,y,r)$ if and only if $D(y,x,r)$.

3.
For all $x,y\in X$ and all $r,s\in {\mathbb{Q}}_{+}$, if $r\le s$ and $D(x,y,r)$ then $D(x,y,s)$.

4.
For all $x,y,z\in X$ and all $r,s\in {\mathbb{Q}}_{+}$, if $D(x,y,r)$ and $D(y,z,s)$, then $D(x,z,r+s)$.
2 Equivalence with usual definition
The definition presented above is equivalent^{} to the definition presented in metric space (http://planetmath.org/MetricSpace). This equivalence is given by the following equations which relate the primitive term “$D$” used in this definition with the primitive term “$d$” of the previous definition:
$$d(x,y)=inf\{r\mid D(x,y,r)\}$$ 
$$D(x,y,r)\leftrightarrow d(x,y)\le r$$ 
Title  alternative definition of metric space 

Canonical name  AlternativeDefinitionOfMetricSpace 
Date of creation  20130322 15:09:19 
Last modified on  20130322 15:09:19 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  8 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 54E35 