A semi-metric space is a generalization of a metric space where some of the conditions of a metric are relaxed. Specifically, a semi-metric only requires the non-negativity, identity of indiscernibles, and triangle inequality, but not necessarily symmetry. This means that the distance between two points can be different depending on the order.

Here’s an example of a semi-metric:

Define $d : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ by:

$d(x, y) = |x - y|^2$

Reason why this is a semi-metric but not a metric:

1. Non-negativity: $d(x, y) = |x - y|^2 \geq 0$ for all $x, y$, so it satisfies non-negativity.
2. Identity of indiscernibles: If $x = y$, then $d(x, x) = |x - x|^2 = 0$. If $x \neq y$, $d(x, y) > 0$, so it satisfies the identity of indiscernibles.
3. Triangle inequality: This can be proven since $|x - y|^2$ still respects the triangle inequality property.
4. Symmetry: However, symmetry is not required in a semi-metric, but in this case, it holds true for $d(x, y) = |x - y|^2$.

A less trivial example without symmetry:

Define another function $d : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$:

$d(x, y) = x^2 - y^2 \text{ for } x \geq y \quad \text{and} \quad d(x, y) = 0 \text{ for } x < y$

This is a semi-metric because:

1. Non-negativity: $d(x, y) \geq 0$ for all $x \geq y$, as $x^2 - y^2 \geq 0$, and $d(x, y) = 0$ for $x < y$.
2. Identity of indiscernibles: $d(x, x) = 0$, and $d(x, y) > 0$ if $x > y$, so it satisfies this condition.
3. Triangle inequality: This can be verified for specific cases.
4. Symmetry: Symmetry is not satisfied because $d(x, y) \neq d(y, x)$ in general.

Thus, this is a semi-metric space, but not a metric.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the exact differences between a semi-metric space and a metric space?
2. Can a semi-metric space still be used in practical applications like a metric space?
3. How can we prove the triangle inequality in a semi-metric space?
4. What other generalizations of metric spaces exist besides semi-metrics?
5. How does the relaxation of symmetry affect the properties of the space?

Tip: In a semi-metric, distances can depend on the order of points, so always test the symmetry condition to distinguish it from a metric space!