上半平面
H = H + = H + = { z ∈ C ∣ I z > 0 } = { x + y i ∣ x , y ∈ R , y > 0 } , {\displaystyle {\mathfrak {H}}={\mathfrak {H}}^{+}={\mathfrak {H}}_{+}=\{z\in \mathbb {C} \mid \Im \,z>0\}=\{x+yi\mid x,y\in \mathbb {R} ,\,y>0\},}
H ̄ = H ∪ ∂ H = H + ∪ R ∪ { ∞ } . {\displaystyle {\bar {\mathfrak {H}}}={\mathfrak {H}}\cup \partial {\mathfrak {H}}={\mathfrak {H}}^{+}\cup \mathbb {R} \cup \{\infty \}.}
H ̄ ∪ H − = H + ∪ H − ∪ R ∪ { ∞ } = Riemann sphere . {\displaystyle {\bar {\mathfrak {H}}}\cup {\mathfrak {H}}_{-}={\mathfrak {H}}_{+}\cup {\mathfrak {H}}_{-}\cup \mathbb {R} \cup \{\infty \}={\mbox{ Riemann sphere}}.}