# nLab non-Hausdorff manifold

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

A manifold is usually by default assumed to have an underlying topological space which is a Hausdorff topological space. If that condition is explicitly dropped, one accordingly speaks of a non-Hausdorff manifold. Compare the red herring principle.

## Properties

Assuming that we're talking about topological manifolds, a non-Hausdorff manifold is still T-1? and sober.

## Examples

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

## 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 November 13, 2012 06:22:22 by Toby Bartels (98.23.144.251)