Understanding the Triangular Inequality in a Metric Space

In mathematics, particularly in real analysis and topology, a metric space is a set where a concept of distance between elements of the set is defined. This distance is given by a function called a metric or distance function.

A set $E$ equipped with a function $d: E \times E \to \mathbb{R}$ is called a metric space $(E, d)$ if for all $x, y, z \in E$, the function $d$ satisfies the following properties (axioms of a metric):

• Non-negativity: $d(x, y) \ge 0$
• Identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$
• Symmetry: $d(x, y) = d(y, x)$
• Triangular Inequality: $d(x, y) \le d(x, z) + d(z, y)$

The Triangular Inequality Explained

The fourth property listed above is the triangular inequality. It is a fundamental axiom of a metric space. This property formalizes the intuitive idea that the shortest distance between two points is a straight line. If you want to go from point $x$ to point $y$, taking a detour through a third point $z$ will always result in a distance that is greater than or equal to the direct distance from $x$ to $y$.

Mathematically, the triangular inequality states that for any three points $x, y, z$ in the metric space $(E, d)$, the distance between $x$ and $y$, denoted by $d(x, y)$, is less than or equal to the sum of the distance between $x$ and $z$, $d(x, z)$, and the distance between $z$ and $y$, $d(z, y)$.

So, the correct mathematical expression for the triangular inequality for a distance function $d$ in a metric space is:

$\qquad d(x, y) \le d(x, z) + d(z, y)$

Comparing with the Options

Let's look at the provided options and compare them to the definition of the triangular inequality:

1. $d(x, y) \ne d(x, z) + d(z, y)$: This states the distance is not equal, which doesn't capture the "less than or equal to" nature.
2. $d(x, y) \ge d(x, z) + d(z, y)$: This suggests that taking a detour is always shorter or equal to the direct path, which is incorrect.
3. $d(x, y) \le d(x, z) + d(z, y)$: This precisely matches the definition of the triangular inequality, stating the direct distance is less than or equal to the distance through an intermediate point. This is a crucial axiom for any valid metric.
4. none of these: This would be true only if none of the above options correctly represent the inequality.

Based on the definition of a metric space and its axioms, the third option correctly represents the triangular inequality property of the distance function $d$.