nLab
non-Hausdorff manifold

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