Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space.
The definition in metric space is quite general which has only one axiom(every 3 points collinear) but has a problem: it indeed satisfied by real line, but line segment and ray are also satisfies this condition in themselves when be considered as a metric subspace.
So should an extra axiom be added in? It is $\forall x\forall r \in \mathbb R \exists y,z(d(y,x)+d(x,z)=d(y,z)\land d(y,x)>r \land d(x,z)>r)$ and can make sure every line is infinite long in both two directions.
Besides, is it enough to add this axiom only? e.g. line should be linear continuum.