nLab
non-Hausdorff manifold

Context

Manifolds and cobordisms

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

Idea

It may superficially seem that every locally Euclidean space is a Hausdorff topological space, since Euclidean space is. While the analogous conclusion is in fact true for genuine “local” properties of Euclidean space (such as the T 1T_1-separation axiom, sobriety and local compactness, see this prop.), it is actually wrong for the “non-local” T 2T_2= Hausdorff-conditon: There are locally Euclidean spaces which are not Hausdorff topological spaces, such as the line with two origins.

Since a (paracompact/sigma-compact) Hausdorff locally Euclidean space is called a topological manifold, (paracompact/sigma-compact) non-Hausdorff locally Euclidean spaces are sometimes referred to as “non-Hausdorff manifolds” (an instance of the red herring principle).

Examples

The usual example is the line with two origins: the real line with the point 00 ‘doubled’. Explicitly, this is a quotient space of ×{a,b}\mathbb{R} \times \{a,b\} (given the product topology and with {a,b}\{a,b\} given the discrete topology) by the equivalence relation generated by identifying (x,a)(x,a) with (y,b)(y,b) iff x=y0x = y \ne 0.

References

  • Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity (arXiv:0609098)

  • Steven L. Kent, Roy A. Mimna, and Jamal K. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, (web)

Revised on May 18, 2017 10:11:17 by Urs Schreiber (131.220.184.222)