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The four required properties are: 1. Non-negativity ($d(x,y) \ge 0$), 2.

Conditions: $x, y, z$ are arbitrary elements of the set $A$.; $d$ is the distance function of the metric space.

The four required properties are: 1. Non-negativity ($d(x,y) \ge 0$), 2.

Conditions: $x, y, z$ are arbitrary elements of the set $A$.; $d$ is the distance function of the metric space.

A metric space is formally defined as an ordered pair $(A, d)$, where $A$ is a non-empty set and $d$ is a distance function mapping pairs of elements from $A$ to the real numbers $\mathbb{R}$. The function $d$ must satisfy four specific properties for all elements in $A$.

Conditions: $A$ is a non-empty set.; $d$ is a function $d: A \times A \rightarrow \mathbb{R}$.; The four properties of the distance function must hold.

A metric space is formally defined as an ordered pair $(A, d)$, where $A$ is a non-empty set and $d$ is a distance function mapping pairs of elements from $A$ to the real numbers $\mathbb{R}$. The function $d$ must satisfy four specific properties for all elements in $A$.

Conditions: $A$ is a non-empty set.; $d$ is a function $d: A \times A \rightarrow \mathbb{R}$.; The four properties of the distance function must hold.