Hedgehog space

In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point.

For any cardinal number ${\displaystyle \kappa }$, the ${\displaystyle \kappa }$-hedgehog space is formed by taking the disjoint union of ${\displaystyle \kappa }$ real unit intervals identified at the origin (though its topology is not the quotient topology, but that defined by the metric below). Each unit interval is referred to as one of the hedgehog's spines. A ${\displaystyle \kappa }$-hedgehog space is sometimes called a hedgehog space of spininess ${\displaystyle \kappa }$.

The hedgehog space is a metric space, when endowed with the hedgehog metric ${\displaystyle d(x,y)=\left|x-y\right|}$ if ${\displaystyle x}$ and ${\displaystyle y}$ lie in the same spine, and by ${\displaystyle d(x,y)=\left|x\right|+\left|y\right|}$ if ${\displaystyle x}$ and ${\displaystyle y}$ lie in different spines. Although their disjoint union makes the origins of the intervals distinct, the metric makes them equivalent by assigning them 0 distance.

Hedgehog spaces are examples of real trees.^[1]