Skip to main content

Section 2.3 Metric spaces

Abstract topologies can be very... abstract. The concept of topology, though, was actually inspired by a much more intuitive structure, called metric space, that generalizes what one sees on the real line.
Definition 2.3.1. Metric space.
A metric space is a set \(M\) endowed with a function \(d:M\times M\to[0,\infty)\) such that:
  1. \(d(x,y)=0\) if and only if \(x=y\)
  2. \(d(x,y)=d(y,x)\) for all \(x,y\in M\text{;}\)
  3. \(d(x,z)\leq d(x,y)+d(y,z)\) for all \(x,y,z\in M\)
Of course \(d\) is a generalization of the function \(d(x,y)=|x-y|\text{.}\)

The set \(B_\eps(a)=\{b\in A: d(a,b) < \eps\}\) is called an \(\eps\)-ball centered at \(a\text{.}\)

Each metric space \((M,d)\) is a topological space with the topology \(\cU\) whose open sets are characterized by the following property: \(U\in\cU\) if and only if, for every \(a\in A\text{,}\) there is an \(\eps\)-ball centered at \(a\) entirely contained in \(U\) (see Exercise 2.7.2). With respect to this topology, the distance function is continuous and its continuity is a direct consequence of the triangle inequality (see Exercise 2.7.3).

In a metric space, checking the continuity of a function is much easier than in a general topological space: A similar fact holds for converging sequences: