Metric

The most fundamental idea in single variable calculus, on which other ideas such as derivatives and integrals depend, is the notion of a limit. A limit is an expression of tolerance, which conditions must one impose in order to be within a certain distance of the limit. As such, one needs a notion of distance to define such ideas.

On the real number line this is easy, the distance between $x$ and $y$ is $|x-y|$.

The idea of a metric is to work out which are the fundamental properties of a distance function that make it useful for the sake of limits, and generalise it so that these ideas from calculus can be applied to more exotic sets.

In fact, the first property is not necessary in the definition because

yet it is often included because it is a key part of the intuition of what a metric should be.

There is a close relationship between the definition of a metric and a vector space norm.