Exercise 5, § 1

In Exercise 4, one takes $A =\mathbf{C}[X,Y]$ and I = AX. The space $\mathsf{X}(A)$ is identified with $\mathbf{C}^2$, S $-\{0\}$ is identified with $\{0\} \times \mathbf{C}$. Let U be the set of $(\xi , \eta )\in \mathbf{C}^2$ such that $|\xi |< e^{-|\eta|}$. Prove that $\mathsf{X}'(h)(U)$ is not a neighbourhood of 0 in $\mathsf{X}'(I)$. Deduce that $\mathsf{X}'(I)$ is not identified with the quotient space of $\mathsf{X}'(A)$ by the equivalence relation defined by $\mathsf{X}'(h)$. (On this subject, cf. I, p. 169, Exercise 17.)