Definition:Torus (Topology)/Formal Construction

Definition

Let $T$ be the square embedded in the Cartesian plane defined as:

$T = \closedint {-1} 1 \times \closedint {-1} 1$

Let $T'$ be the quotient space formed from $T$ using the identification mapping $p: T \to T'$ as follows:

$\forall \tuple {x, y} \in T: \map p {x, y} = \begin {cases} \paren {x, y} & : -1 < x, y < 1 \\ \\ \tuple {-x, y} & : x = \pm 1 \\ \\ \tuple {x, -y} & : y = \pm 1 \end {cases}$

Then $T'$ is a torus .

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): torus (anchor ring)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): torus (anchor ring)