Metric space

The guiding example is \(\mathbb{R}\) with \(d(x,y)=|x-y|\), and in \(\mathbb{R}^n\) the Euclidean distance \(d_2(P,Q)=\sqrt{\sum_i (p_i-q_i)^2}\). But the definition does not require the points to be numbers: they can be functions, words or sequences, as long as the distance satisfies the three axioms.

The axioms read as follows: distance is never negative and vanishes only between a point and itself (separation); going from \(x\) to \(y\) costs the same as from \(y\) to \(x\) (symmetry); and a detour through \(y\) never shortens the path from \(x\) to \(z\) (triangle inequality). The last one fails most often: \((x-y)^2\) is symmetric and separates points, but for \(0,1,2\) it gives \(4>1+1\).

One set admits many metrics. On \(\mathbb{R}^2\) the Euclidean, the taxicab \(d_1=|p_1-q_1|+|p_2-q_2|\) and the max distance \(d_\infty\) coexist, and always \(d_\infty\le d_2\le d_1\). There is also the discrete metric, with \(d(x,y)=1\) if \(x\ne y\), which makes all points equidistant. Every subset of a metric space inherits the restricted metric.

With distance in hand we can define the open ball, and from it open sets, limits of sequences and continuity. That is the path this topic follows toward general topology.