Real Line - As A Metric Space

As A Metric Space

The real line forms a metric space, with the metric given by absolute difference:

d(x, y)  =  | x − y | .

If p ∈ R and ε > 0, then the ε-ball in R centered at p is simply the open interval (p − ε, p + ε).

This real line has several important properties as a metric space:

• The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.
• The real line is path-connected, and is one of the simplest examples of a geodesic metric space
• The Hausdorff dimension of the real line is equal to one.
• The isometry group of the real line, also known as the Euclidean group E(1), consists of all functions of the form x ↦ t ± x, where t is a real number. This group is isomorphic to a semidirect product of the additive group of R with a cyclic group of order two, and is an example of a generalized dihedral group.