# Hedgehog space

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{{Short description|Topological space made of a set of spines joined at a point}}
thumb|upright=1.2|A hedgehog space with a large but finite number of spines
In [mathematics](/source/mathematics), a '''hedgehog space''' is a [topological space](/source/topological_space) consisting of a set of spines joined at a point.

For any [cardinal number](/source/cardinal_number) <math>\kappa</math>, the <math>\kappa</math>-hedgehog space is formed by taking the [disjoint union](/source/disjoint_union) of <math>\kappa</math> [real](/source/real_number) [unit interval](/source/unit_interval)s identified at the origin (though its [topology](/source/topological_space) is not the [quotient topology](/source/quotient_topology), but that defined by the [metric](/source/metric_(mathematics)) below). Each unit interval is referred to as one of the hedgehog's ''spines.'' A <math>\kappa</math>-hedgehog space is sometimes called a '''hedgehog space of spininess <math>\kappa</math>'''.

The hedgehog space is a [metric space](/source/metric_space), when endowed with the '''hedgehog metric''' <math>d(x,y)=\left| x - y \right|</math> if <math>x</math> and <math>y</math> lie in the same spine, and by <math>d(x,y)=\left|x\right| + \left|y\right|</math> if <math>x</math> and <math>y</math> 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 tree](/source/real_tree)s.<ref name="carlisle">{{cite conference |first=Sylvia |last=Carlisle |title=Model Theory of Real Trees |conference=Graduate Student Conference in Logic |place=University of Illinois, Chicago, IL |year=2007}}</ref>

==Paris metric==
<!-- This section is linked from [French railroad metric](/source/French_railroad_metric) -->
The metric on the [plane](/source/plane_(geometry)) in which the distance between any two points is their [Euclidean distance](/source/Euclidean_distance) when the two points belong to a [ray](/source/Line_(mathematics)) through the origin, and is otherwise the sum of the distances of the two points from the origin, is sometimes called the '''Paris metric'''<ref name="carlisle"/> because navigation in this metric resembles that in the radial street plan of [Paris](/source/Paris): for almost all pairs of points, the shortest path passes through the center. The Paris metric, restricted to the [unit disk](/source/unit_disk), is a hedgehog space where ''K'' is the [cardinality of the continuum](/source/cardinality_of_the_continuum).

==Kowalsky's theorem==
Kowalsky's theorem, named after Hans-Joachim Kowalsky,<ref>{{cite book |first=H.J. |last=Kowalsky |title=Topologische Räume |trans-title=Topological Spaces |lang=de |publisher=Birkhäuser |place=Basel-Stuttgart |year=1961}}</ref><ref>{{cite journal |first=M.A. |last=Swardson |title=A short proof of Kowalsky's hedgehog theorem |journal=[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society) |volume=75 |year=1979 |issue=1 |page=188 |doi=10.1090/s0002-9939-1979-0529240-7 |doi-access=free}}</ref> states that any [metrizable space](/source/metrizable_space) of [weight](/source/weight_of_a_topological_space) <math>\kappa</math> can be represented as a [topological subspace](/source/topological_subspace) of the [product](/source/product_topology) of [countably many](/source/countable_set) {{nowrap|<math>\kappa</math>-hedgehog}} spaces.

== See also ==
* [Comb space](/source/Comb_space)
* [Long line (topology)](/source/Long_line_(topology))
* [Rose (topology)](/source/Rose_(topology))

==References==
{{reflist|25em}}

==Other sources==
*{{cite book |first1=A.V. |last1=Arkhangelskii |author-link=Alexander Arhangelskii |first2=L.S. |last2=Pontryagin |author-link2=Lev Pontryagin |title=General Topology |volume=I |year=1990 |publisher=Springer-Verlag |place=Berlin, DE |isbn=3-540-18178-4}}
*{{cite book |first1=L.A. |last1=Steen |first2=J.A. Jr. |last2=Seebach |title=Counter-Examples in Topology |year=1970 |publisher=Holt, Rinehart, and Winston}}
*{{cite arXiv |last1=Torres |first1=Igor |title=A tale of three hedgehogs |date=2017 |class=math.GN |eprint=1711.08656 }}

Category:General topology
Category:Topological spaces
Category:Trees (topology)

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Adapted from the Wikipedia article [Hedgehog space](https://en.wikipedia.org/wiki/Hedgehog_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hedgehog_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.