real projective space

实射影空间

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a nonnegative integer n, the space of one-dimensional real linear subspaces of real (n+1)-space, topologized as the quotient of nonzero vectors by multiplication by nonzero real scalars.

对非负整数 $n$，实 $(n+1)$ 维空间中一维线性子空间所成的空间；用非零向量在非零实数倍关系下的商拓扑。

Example

The points $[1:2:3]$ and $[-2:-4:-6]$ in $\mathbb{RP}^2$ coincide. Equivalently, $\mathbb{RP}^2$ is the quotient of $S^2$ identifying antipodal points.

Usage and distinctions

The dimension is n despite using n+1 homogeneous coordinates. This is a topological quotient over the reals, distinct from a complex projective space or an algebraic scheme construction.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic TopologyChapter 0, Cell Complexes, Example 0.4, printed p. 6 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第7周10月26日，主要概念：实射影空间 · Checked 2026-10-14

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