Hedgehog space

A hedgehog space with a large but finite number of spokes

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

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: for almost all pairs of points, the shortest path passes through the center. The Paris metric, restricted to the unit disk, is a hedgehog space where K is the cardinality of the continuum.

Kowalsky's theorem

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

• Comb space
• Long line (topology)
• Rose (topology)

Notes

1. Carlisle (2007).
2. Kowalsky (1961); Swardson (1979).

References

• 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.