Example 2

Soit $n$ an integer $\geqslant 0$. Let $P_n$ denote the set of monic polynomials $P\in$ $\mathbf{C}[X]$ of degree $n$, endowed with the topology for which the mapping $(c_0, . . . , c_{n-1})\mapsto X^n+c_{n-1}X^{n-1}+\cdots +c_0$ is a homeomorphism of $\mathbf{C}^n$ onto $P_n$. The mapping $p$ of $\mathbf{C}^n$ into $P_n$ defined by $p(z_1, . . . , z_n) = (X-z_1). . .(X-z_n)$ is continuous. The symmetric group $\mathfrak{S}_n$ operates on $\mathbf{C}^n$ by permutation of the factors and $p$ defines by passing to the quotient a homeomorphism of $\mathbf{C}^n/\mathfrak{S}_n$ onto $P_n$ (TG, VIII, p. 22, prop. 1, I, p. 23, cor. 1 and TG, VIII, p. 20). We therefore deduce the following statement:

Let I be an interval of $\mathbf{R}$, let $(c_0, . . . , c_{n-1})$ be a sequence of continuous mappings from I into $\mathbf{C}$, let $a$ be a point of I and let $(z_1, . . . , z_n)$ be a sequence of complex numbers such that one has $(X-z_1). . .(X-z_n) = X^n+$ $c_{n-1}(a)X^{n-1}+\cdots +c_0(a)$. There exists a sequence $(\lambda_1, . . . , \lambda_n)$ of continuous mappings from I into $\mathbf{C}$ such that one has $\lambda_i(a) =z_i$ for $1\leqslant i\leqslant n$ and $(X-\lambda_1(t)). . .(X-\lambda_n(t)) = X^n+c_{n-1}(t)X^{n-1}+\cdots +c_0(t)$ for every $t\in I$.