Compactness and Liouville distinguish the three models

The sphere $\widehat{\mathbb{C}}$ is compact: $\mathrm{\Sigma }$ is a homeomorphism onto $S^2$ [F6], the sphere $S^2\subseteq {\mathbb{R}}^3$ is compact [F4], and ${\mathrm{\Sigma }}^{-1}$ is continuous, so $\widehat{\mathbb{C}}={\mathrm{\Sigma }}^{-1}(S^2)$ is a continuous image of a compact set [F5].

Neither $\mathbb{C}$ nor $\mathbb{D}$ is compact. The open discs $D(0,n)$, $n\ge 1$, cover $\mathbb{C}$; any finitely many of them are contained in $D(0,N)$ for $N$ the largest index occurring, which omits every point of modulus greater than $N$, so no finite subfamily covers $\mathbb{C}$. Likewise the open discs $D(0,1-1\mathrm{/}n)$, $n\ge 2$, cover $\mathbb{D}$; any finitely many are contained in $D(0,1-1\mathrm{/}N)$ for $N$ the largest index occurring, which omits the points of modulus between $1-1\mathrm{/}N$ and $1$, so no finite subfamily covers $\mathbb{D}$. By the definition of compactness neither space is compact.

The plane is not biholomorphic to the disc: if $f:\mathbb{C}\to \mathbb{D}$ were a biholomorphism, then by [F8] $f$ is holomorphic and bijective, and regarding it as a map into $\mathbb{C}$ it is entire with $\mathrm{\mid }f(z)\mathrm{\mid }<1$ for every $z$; by [F2] such an $f$ must be constant, and a constant map is not injective, hence not bijective, a contradiction. So no biholomorphism $\mathbb{C}\to \mathbb{D}$ exists.

Suppose there were a biholomorphism $f:\widehat{\mathbb{C}}\to \mathbb{C}$; by the given meaning of biholomorphism and [F7], both it and its inverse are continuous, so it is a homeomorphism and is surjective onto $\mathbb{C}$. Since $\widehat{\mathbb{C}}$ is compact by step 1.1, its continuous image $\mathbb{C}$ would be compact [F5], contradicting step 1.2. The same argument with $\mathbb{D}$ in place of $\mathbb{C}$ excludes a biholomorphism $\widehat{\mathbb{C}}\to \mathbb{D}$. So compactness separates the sphere from the plane and the disc.

The obstruction in step 1.3 is not topological. The map $\mathrm{\Phi }(z):=z\mathrm{/}(1+\mathrm{\mid }z\mathrm{\mid })$ is a continuous bijection of $\mathbb{C}$ onto $\mathbb{D}$, because $\mathrm{\mid }\mathrm{\Phi }(z)\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{/}(1+\mathrm{\mid }z\mathrm{\mid })<1$ with equality approached but never attained, and its inverse is ${\mathrm{\Phi }}^{-1}(w)=w\mathrm{/}(1-\mathrm{\mid }w\mathrm{\mid })$, also continuous; so $\mathbb{C}$ and $\mathbb{D}$ are homeomorphic. Nevertheless step 1.3 shows they are not biholomorphic, while [F1] independently records the pairwise non-bihomorphism of all three models. Hence the two distinctions exhibited above — compactness for the sphere, Liouville for the plane versus the disc — are the classical witnesses for the inequivalence of the three simply connected models. Every cover and every map used is given by an explicit formula, so no choice principle is used.