Non-Hausdorff manifold

In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.

Examples

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Line with two origins

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The most familiar non-Hausdorff manifold is the line with two origins,[1] or bug-eyed line. This is the quotient space of two copies of the real line,   and   (with  ), obtained by identifying points   and   whenever  

An equivalent description of the space is to take the real line   and replace the origin   with two origins   and   The subspace   retains its usual Euclidean topology. And a local base of open neighborhoods at each origin   is formed by the sets   with   an open neighborhood of   in  

For each origin   the subspace obtained from   by replacing   with   is an open neighborhood of   homeomorphic to  [1] Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of   intersects every neighbourhood of   It is however a T1 space.

The space is second countable.

The space exhibits several phenomena that do not happen in Hausdorff spaces:

  • The space is path connected but not arc connected. In particular, to get a path from one origin to the other one can first move left from   to   within the line through the first origin, and then move back to the right from   to   within the line through the second origin. But it is impossible to join the two origins with an arc, which is an injective path; intuitively, if one moves first to the left, one has to eventually backtrack and move back to the right.
  • The intersection of two compact sets need not be compact. For example, the sets   and   are compact, but their intersection   is not.
  • The space is locally compact in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure. So the space is not a regular space, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.

The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.[2]

Line with many origins

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The line with many origins[3] is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set   with the discrete topology and taking the quotient space of   that identifies points   and   whenever   Equivalently, it can be obtained from   by replacing the origin   with many origins   one for each   The neighborhoods of each origin are described as in the two origin case.

If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set   is the set   obtained by adding all the origins to  , and that closure is not compact. From being locally Euclidean, such a space is locally compact in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.

Branching line

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Similar to the line with two origins is the branching line.

This is the quotient space of two copies of the real line   with the equivalence relation  

This space has a single point for each negative real number   and two points   for every non-negative number: it has a "fork" at zero.

Etale space

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The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)[4]

Properties

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Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable in general) and locally Hausdorff (but not Hausdorff in general).

See also

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Notes

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  1. ^ a b Munkres 2000, p. 227.
  2. ^ Gabard 2006, Proposition 5.1.
  3. ^ Lee 2011, Problem 4-22, p. 125.
  4. ^ Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer-Verlag. p. 164. ISBN 978-0-387-90894-6.

References

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