# 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 ${\displaystyle K}$, the ${\displaystyle K}$-hedgehog space is formed by taking the disjoint union of ${\displaystyle K}$ real unit intervals identified at the origin. Each unit interval is referred to as one of the hedgehog's spines. A ${\displaystyle K}$-hedgehog space is sometimes called a hedgehog space of spininess ${\displaystyle K}$.

The hedgehog space is a metric space, when endowed with the hedgehog metric ${\displaystyle d(x,y)=|x-y|}$ if ${\displaystyle x}$ and ${\displaystyle y}$ lie in the same spine, and by ${\displaystyle d(x,y)=x+y}$ if ${\displaystyle x}$ and ${\displaystyle y}$ 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 metrizable space of weight ${\displaystyle K}$ can be represented as a topological subspace of the product of countably many ${\displaystyle K}$-hedgehog spaces.