TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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Stereographic projection identifies the Riemann sphere with the unit two-sphere

Statement

Let $S^{2} : = {(x, y, t) \in R^{3} : x^{2} + y^{2} + t^{2} = 1} .$ Define $Σ (z) = (\frac{2 Re z}{1 + ∣ z ∣^{2}}, \frac{2 Im z}{1 + ∣ z ∣^{2}}, \frac{∣ z ∣^{2} - 1}{1 + ∣ z ∣^{2}}) (z \in C)$ and $Σ (\infty) = (0, 0, 1)$ . Then $Σ : \hat{C} \to S^{2}$ is a homeomorphism, with inverse $Π (x, y, t) = {\begin{cases} \frac{x + i y}{1 - t}, & (x, y, t) \neq (0, 0, 1), \\ \infty, & (x, y, t) = (0, 0, 1) . \end{cases}$

Facts & Assumptions

Given: The Riemann sphere $\hat{C} = C \cup {\infty}$ , the unit sphere $S^{2}$ , and the displayed formulas for $Σ$ and $Π$ .

[L1]

In the one-point compactification, a neighbourhood of $\infty$ is exactly the complement of a closed compact subset of $C$ , and $\hat{C}$ is compact Hausdorff (The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ , $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

Proof

technique · direct

1.1givenalgebra

Direct algebra gives $Σ (z) \in S^{2}$ for every finite $z$ , and the displayed formulas satisfy $Σ (Π (x, y, t)) = (x, y, t)$ for $(x, y, t) \neq (0, 0, 1)$ and $Π (Σ (z)) = z$ for finite $z$ , with $Σ (\infty) = (0, 0, 1)$ and $Π (0, 0, 1) = \infty$ .

1.2L1givenalgebra

On $C$ and on $S^{2} ∖ {(0, 0, 1)}$ the formulas are rational with nonzero denominator, so both restrictions are continuous; and for the cap $U_{t} = {(x, y, s) : s > t} \cup {(0, 0, 1)}$ one has $Σ^{- 1} (U_{t}) = {z : ∣ z ∣ > \sqrt{(1 + t) / (1 - t)}} \cup {\infty}$ , which is a neighbourhood of $\infty$ by [L1].

1.3L1choosealgebra

If $V = \hat{C} ∖ K$ is a neighbourhood of $\infty$ , compactness of $K$ gives $R > 0$ with $K \subseteq \overline{D} (0, R)$ , so the cap $U_{(R^{2} - 1) / (R^{2} + 1)}$ satisfies $Π (U_{(R^{2} - 1) / (R^{2} + 1)}) \subseteq V$ ; therefore $Π$ is continuous at the north pole.

2.1given∎

The maps $Σ$ and $Π$ are continuous inverse bijections by the preceding three steps, so $Σ$ is a homeomorphism.

Stereographic projection identifies the Riemann sphere with the unit two-sphere

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources