Example 2

The absolute value of a real number (Chapter IV, § 1, no. 6) satisfies axioms (VM_I), (VM_{II}) and (VM_{III}), and the topology it defines on the field $\mathbf{R}$ is the topology of the real line. On the field $\mathbf{C}$ of complex numbers (identified with $\mathbf{R}^2$) [resp. the division ring $\mathbf{H}$ of quaternions (identified with $\mathbf{R}^4$)] the Euclidean norm is again an absolute value and defines the topology of the field $\mathbf{C}$ (resp. the division ring $\mathbf{H}$) (Chapter VIII, § 1, nos. 2 and 4).