In some book I found, a metric on a non-empty set $X$ defined as a map $$X\times X\to \Bbb R^{+}$$ and some other place as $$X\times X\to \Bbb R$$ So, is a metric a real valued function or a non-negative real valued function or it doesn't matter at all?
Since metric is a generalization of distance on the real line. I was convinced by the first definition. But the second one? I don't know and the worst part is, it is widely used definition.
On
You didn't read far enough when you looked at "some other place." If you had, you'd see nonnegativity is included as an axiom. Metrics always take nonnegative values.
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A metric on $X$ is a map $d:X\times X\to \mathbb R$ that satisfies certain conditions:
The first condition implies $d:X\times X\to [0,\infty)$.
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Non-negativity of a metric is implied by the axioms that define it. This can be seen in the following way. Take any $x, y \in X$. Then from the triangle inequality, we have $d(x, y) + d(y, x) \ge d(x, x)$. But from symmetry of the metric, $d(x, y) = d(y, x)$, so that $2d(x, y) \ge d(x, x)$. And from the fact that $d(x, x) = 0$, we get $d(x, y) \ge 0$.
Sometimes $d(x, y) \ge 0$ is taken as an axiom depending on whose definition you use, but all these definitions are equivalent.
If you write $X \times X \to \Bbb R^+$, you don't need to note that $d(x,y) \geq 0$ for all $x,y$, since it is already implied by the use of $\Bbb R^+$. If you write only $X \times X \to \Bbb R$, you must call the attention to this fact.