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.
Read more about this topic: Real Line
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