nLab
plane

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Manifolds and cobordisms

Contents

Definition

The plane is the Cartesian space 2\mathbb{R}^2. This is naturally a topological space, a manifold, and a smooth manifold. If we take one of the axes (traditionally the second) to be imaginary, then this real plane may be identified with the complex plane 1\mathbb{C}^1. As a stage for Euclidean geometry, it may be called the Cartesian plane, Euclidean plane, or coordinate plane.

References

For discussion of the plane via axioms for the points and lines in it (synthetic geometry) see the references at Euclidean geometry.

Last revised on October 3, 2018 at 10:46:13. See the history of this page for a list of all contributions to it.