Straight line axiom In geometry there is a straight line Axiom stated as follows: For all two different points $A,B \in \mathcal{R}$ and for all $\delta >0$ there exists exactly one point $C \in \mathcal{R}$ such that $|BC|=\delta$ and $|AB| + |BC|= |AC|$, where $|\cdot , \cdot |$ is our metric on a nonempty set $\mathcal{R}$ and $\mathcal{R}$ is a nonempty set interpreted as our space. The obvious picture of this axiom is the following

I haven't understood what's preventing me from letting $C$ be the following point in $\mathcal{R}$

I know from a real world point of view this isn't the "right point" but abstractly speaking since $|\cdot , \cdot |$ is just a function with three properties can't i just define it however I want?
 A: There are two important things to distinguish. One is the traditional Euclidean plane where you have certain expectations of what a straight line actually is. In that view, the point in your image would not be the correct one, and the reason for that is that the well-established Euclidean metric says otherwise.
If you redefine that metric, then a lot of intuition breaks down. There is nothing saying your point $C$ can't be where you put it, as long as the shortest route from there to $A$ still goes through $B$.
The sharp bend in your diagram might appear a bit unmotivated. But you can actually get some physical setup where you would get that behaviour. Light rays follow the shortest path in terms of the time it takes to travel. That doesn't have to be the geometrically shortest distance if it travels through different materials where the speed of light is different. So at the interface between two materials you would get refraction resulting in a bend like in your picture. And the underlying time-based metric should be compatible with whatever properties you expect a metric to have. By the definition you have, such a ray of light would be considered "straight" according to the time-based metric, although it wouldn't be according to conventional distance-based metric.
