“Non-Hausdorff manifold”的意思、由来-开放百科全书

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Non-Hausdorff manifold

释义

Examples
Line with two origins Branching line Etale space Quantum path integration (sum over histories)
Notes
References

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

Line with two origins

The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line.

This is the quotient space of two copies of the real line

R × {a} and R × {b}

with the equivalence relation

This space has a single point for each nonzero real number r and two points 0_a and 0_b. A local base of open neighborhoods of in this space can be thought to consist of sets of the form , where is any positive real number. A similar description of a local base of open neighborhoods of is possible. Thus, in this space all neighbourhoods of 0_a intersect all neighbourhoods of 0_b, so it is non-Hausdorff.

Further, the line with two origins does not have the homotopy type of a CW-complex, or of any Hausdorff space.^[1]

Branching line

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

This is the quotient space of two copies of the real line

R × {a} and R × {b}

with the equivalence relation

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

Etale space

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.){{Citation needed|date=July 2010}}

Quantum path integration (sum over histories)

Mark F. Sharlow claims that functional spacetime of path integral has a "branching" non-Hausdorff structure.^[2]

Notes

1. ^Gabard, pp. 4–5
2. ^{{cite web|author1=Mark F. Sharlow|title=The quantum mechanical path integral: toward a realistic interpretation|url=https://www.eskimo.com/~msharlow/path.pdf|date=2005–2007}}

References

{{citation

|title=Manifolds: Hausdorffness versus homogeneity
|first=Mathieu
|last=Baillif
|first2=Alexandre
|last2=Gabard
|arxiv=math.GN/0609098v1
}}

{{citation

|title=A separable manifold failing to have the homotopy type of a CW-complex
|first=Alexandre
|last=Gabard
|arxiv=math.GT/0609665v1
}}

3 : Manifolds|Topology|General topology

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