A **semi-metric space** is a generalization of a metric space where the distance function, or semi-metric, satisfies all the properties of a metric except for the triangle inequality. Specifically, a semi-metric space is defined by a set \( X \) along with a function \( d: X \times X \rightarrow \mathbb{R} \) such that 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**: \( d(x, y) = d(y, x) \).

However, unlike a metric space, the triangle inequality may not hold in a semi-metric space:

\[ d(x, z) \leq d(x, y) + d(y, z) \]

### Example of a Semi-Metric Space

Consider the set \( X = \{a, b, c\} \) with a semi-metric defined by the following distance function \( d \):

- \( d(a, a) = d(b, b) = d(c, c) = 0 \)
- \( d(a, b) = 1 \)
- \( d(a, c) = 2 \)
- \( d(b, c) = 3 \)

Here, the distance function \( d \) satisfies non-negativity, identity of indiscernibles, and symmetry. However, it does

Explore the concept of semi-metric spaces in mathematics, defined by a distance function that satisfies non-negativity, identity of indiscernibles, and symmetry but may not satisfy the triangle inequality. Example problems and applications included.

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