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. Although their disjoint union makes the origins of the intervals distinct, the metric identifies them by assigning them 0 distance.

Hedgehog spaces are examples of real trees.[1]

Paris metric[edit]

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[1] 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.

Kowalsky's theorem[edit]

Kowalsky's theorem, named after Hans-Joachim Kowalsky,[2] states that any metric space of weight K can be represented as a topological subspace of the product of countably many K-hedgehog spaces.

See also[edit]

Notes[edit]

References[edit]

  • Arkhangelskii, A. V.; Pontryagin, L. S. (1990), General Topology I, Berlin: Springer-Verlag, ISBN 3-540-18178-4 .
  • Carlisle, Sylvia (2007), "Model Theory of Real Trees", Graduate Student Conference in Logic, Univ. of Illinois, Chicago .
  • Kowalsky, H. J. (1961), Topologische Räume, Basel-Stuttgart: Birkhäuser .
  • Steen, L. A.; Seebach, J. A., Jr. (1970), Counterexamples in Topology, Holt, Rinehart and Winston .
  • Swardson, M. A. (1979), "A short proof of Kowalsky's hedgehog theorem", Proc. Amer. Math. Soc. 75 (1): 188, doi:10.1090/s0002-9939-1979-0529240-7 .