Hedgehog Space
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In mathematics, a hedgehog space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
consisting of a set of spines joined at a point. For any
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
\kappa, the \kappa-hedgehog space is formed by taking the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
of \kappa
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analys ...
s identified at the origin (though its
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
is not the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
, but that defined by the metric below). Each unit interval is referred to as one of the hedgehog's ''spines.'' A \kappa-hedgehog space is sometimes called a hedgehog space of spininess \kappa. The hedgehog space is a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
, when endowed with the hedgehog metric d(x,y)=\left, x - y \ if x and y lie in the same spine, and by d(x,y)=\left, x\ + \left, y\ if x and 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.


Paris metric

The metric on the plane in which the distance between any two points is their
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
when the two points belong to a
ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (g ...
though the origin, and is otherwise the sum of the distances of the two points from the origin, is sometimes called the Paris metric because navigation in this metric resembles that in the radial street plan of
Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 kmĀ² (41 sq mi), making it the 30th most densely populated city in the world in 2020. ...
: for almost all pairs of points, the shortest path passes through the center. The Paris metric, restricted to the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...
, is a hedgehog space where ''K'' is the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...
.


Kowalsky's theorem

Kowalsky's theorem, named after Hans-Joachim Kowalsky, states that any
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty ...
of
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar q ...
\kappa can be represented as a
topological subspace In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
of the product of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
spaces.


See also

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


References


Other sources

* * *{{cite arXiv , last1=Torres , first1=Igor , title=A tale of three hedgehogs , date=2017 , class=math.GN , eprint=1711.08656 General topology Topological spaces Trees (topology)