Example 3

An etale morphism of varieties (VAR, R, 5.7.8) is a topologically etale mapping, but there exist morphisms of real varieties which are topologically etale and which are not etale morphisms. This is the case, for example, of the mapping $x\mapsto x^3$ of $\mathbf{R}$ into $\mathbf{R}$. However, a morphism of complex analytic varieties which is topologically etale is an etale morphism (I, p. 141, exerc. 6).