DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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The two-dimensional torus $T^{2} = (R / Z)^{2}$

Definition

Let $Q = R / Z$ be the quotient circle of The circle as $S^{1} = R / Z$ with basepoint $[0]$ . The two-dimensional torus is the product space

$T^{2} : = Q \times Q$

with the product topology (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), pointed at $([0], [0])$ . This definition uses the quotient-circle model in both coordinates.

Depends on

Used by

π₁(T²)≅ℤ×ℤ Corollary

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Peter May, A Concise Course in Algebraic Topology, Chapter 2, Section 8 (standard reference, not scraped)

The two-dimensional torus T2=(R/Z)2T2=(R/Z)2