A metric space is a set $X$ along with a function $d: X \times X \to \mathbb{R}$ (called a metric or distance function) that satisfies the following four properties for all $x, y, z \in X$:

1. Non-negativity: $d(x, y) \geq 0$.
2. Identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$.
3. Symmetry: $d(x, y) = d(y, x)$.
4. Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$.

An example of a set and function that do not form a metric space would be one where at least one of these conditions is violated.

Example:

Consider the set $X = \mathbb{R}$ (the set of real numbers) and define the function $d: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ by:

$d(x, y) = |x - y| - 1$

Let's check the properties:

1. Non-negativity: For $x = 0$ and $y = 1$, $d(0, 1) = |0 - 1| - 1 = 0 - 1 = -1$, which is negative. This violates the non-negativity property.

2. Identity of indiscernibles: Suppose $x = y$. Then $d(x, x) = |x - x| - 1 = 0 - 1 = -1$. This violates the property that $d(x, x)$ should be zero.

3. Symmetry: $d(x, y) = |x - y| - 1$ is symmetric because $|x - y| = |y - x|$, so $d(x, y) = d(y, x)$.

4. Triangle inequality: Consider $x = 0$, $y = 1$, and $z = 2$. We have $d(0, 2) = |0 - 2| - 1 = 2 - 1 = 1$ and $d(0, 1) + d(1, 2) = (|0 - 1| - 1) + (|1 - 2| - 1) = 0 + 0 = 0$. However, $d(0, 2) > d(0, 1) + d(1, 2)$, which violates the triangle inequality.

Since $d(x, y) = |x - y| - 1$ fails the non-negativity, identity of indiscernibles, and triangle inequality conditions, $(\mathbb{R}, d)$ is not a metric space.

Would you like more details or have any questions? Here are five related questions:

1. What are other examples of non-metric spaces?
2. Can a metric space be defined on non-real number sets?
3. How does violating each metric property affect the structure of the space?
4. What is the significance of the triangle inequality in metric spaces?
5. How can you modify a non-metric function to satisfy the metric properties?

Tip: When defining a metric, always ensure that the distance function meets all four metric properties to maintain a valid metric space structure.