Hedgehog space
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In mathematics, a hedgehog space is a topological space, consisting of a set of spines joined at a point.
For any cardinal number K, the K-hedgehog space is formed by taking the disjoint union of K real unit intervals identified at the origin. Each unit interval is referred to as one of the hedgehog's spines. A K-hedgehog space is sometimes called a hedgehog space of spininess K.
The hedgehog space is a metric space, when endowed with the hedgehog metric d(x,y) = | x − y | if x and y lie in the same spine, and by d(x,y) = x + y if x and y lie in different spines.
The hedgehog space is an example of a Moore space, and satisfies many strong normality and compactness properties. Hedgehog spaces are examples of real trees.
[edit] Paris metric
The metric on the plane in which the distance between any two points is their Euclidean distance when the two points belong to a ray though the origin, and is otherwise the sum of the distances of the two points from the origin, is sometimes called the Paris metric because navigation in this metric resembles that in the radial street plan of Paris. The Paris metric, restricted to the unit disk, is a hedgehog space where K is the cardinality of the continuum.
[edit] Kowalsky's theorem
Kowalsky's theorem[1] states that any metric space of weight K can be represented as a subspace of the product of countably many K-hedgehog spaces.
[edit] References
- ^ H.J. Kowalsky, Topologische Räume, Birkhäuser, Basel-Stuttgart (1961)
- L.A. Steen, J.A.Seebach, Jr., Counterexamples in Topology, (1970) Holt, Rinehart and Winston, Inc..
- A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4
- This article incorporates material from Hedgehog space on PlanetMath, which is licensed under the GFDL.
