Let $R$ be a rational function of degree $d$ mapping the Riemann sphere to itself:$$R(z) = \frac{a_d z^d + a_{d-1} z^{d-1} + \cdots + a_0}{b_d z^d + b_{d-1} z^{d-1} + \cdots + b_0}.$$where $a_d$ and $b_d$ are not both zero. And suppose that a sequence of coefficients ${(a_d, a_{d-1}, \ldots, a_0; b_d, b_{d-1}, \ldots, b_0)_n}$ converges to the coefficients of $R$. R$in$\mathbb{C}^{2d+2}$. Let$R_n$be the rational function with coefficients$(a_d, a_{d-1}, \ldots, a_0; b_d, b_{d-1}, \ldots, b_0)_n$. I ask because I am reading Beardon's Iteration of Rational Functions and in section 2.8 he introduces a mapping$\Psi:\mathcal{R}\to\mathbb{C}\mathbb{P}^{2d+1}$, which takes a rational function to its coefficients. Beardon omits the details, but claims that$\Psi$is a homeomorphism of$\mathcal{R}_d$onto$\Psi(\mathcal{R}_d)$. Now my \Psi(\mathcal{R}_d)$, where $\mathcal{R}_d$ is the space of rational functions having degree exactly $d$.