What is the first required property of a metric in this video?

It implies that $x$ and $y$ are the same point ($x = y$). This is the identity of indiscernibles property.

Conditions: $x$ and $y$ are points in the metric space.; $d(x,y) = 0$.

The video explains the triangle inequality using a detour analogy. It compares going straight from point $x$ to point $z$ with taking a detour through an intermediate point $y$.

Conditions: $x, y, z$ are points in the metric space.; $y$ is an arbitrary intermediate point.

Distance is independent of direction because of the symmetry property of the metric. The formula $d(x,y) = d(y,x)$ shows that swapping the order of the points does not change the distance value.

Conditions: $x$ and $y$ are points in 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.

The video uses the analogy of commuting from work to home. Going straight home represents the direct distance $d(x,z)$.

Conditions: $x$ is work, $z$ is home, and $y$ is the shop.; The path is a journey in a metric space.