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.Q:

What is the relationship between metric and non-metric?

Among the various types of metrics, one metric has the most generality and can easily be applied to most disciplines. It is the euclidean distance.
However, the euclidean distance is also easily applicable to some algebraic structures (e.g. rings).
For example, if one has a collection of $x_i$, one can calculate the distances between them with the formula
$$d(x_i,x_j)=\sqrt{(x_i-x_j)^T(x_i-x_j)}$$
This formula is easily generalized to the case of $x_i$ being a vector, or matrix, and is called the Mahalanobis distance.
Is there a metric that is just as generic and applicable as the euclidean distance, but that only works with metric spaces?

A:

I would be tempted to say that there is no such metric. If that is the case, why do we use Euclidean distance? We use Euclidean distance as it is simple to derive an intuitive meaning that many people will get.
As a more direct answer to your question, you can take the following definitions and see whether they give you what you want:
Definition 1: A metric on a set $\mathcal{S}$ is a function $\mathcal{M}: \mathcal{S} \times \mathcal{S} \rightarrow \mathbb{R}$ where for all $x, y, z \in \mathcal{S}$:
$\mathcal{M}(x,y) \geq 0$,
$\mathcal{M}(x,y) = 0 \iff x=y$,
$\mathcal{M}(x,y) = \mathcal{M}(y,x)$,
$\mathcal{M}(x,z) \leq \mathcal{M}(x,y) + \mathcal{
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