What is the formal definition of a metric space in this lecture?

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.

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.

The video uses absolute value because simple subtraction can yield a negative result depending on the order of the operands. Distance must be non-negative ($d(x,y) \ge 0$).

Conditions: The points are real numbers on a number line.; The distance function must satisfy the non-negativity property.