Hedgehog space
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 points ...
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. Th ...
\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 (th ...
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) (2010) ...
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 analysis, ...
s identified at the origin (though its
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
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 settin ...
, 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 tree In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the s ...
s.


Paris metric

The metric on the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
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, therefor ...
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 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. S ...
: 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 di ...
, 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 , \mathb ...
.


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 Metric or metrical may refer t ...
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 Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
\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 to ...
of the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
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 number ...
spaces.


See also

*
Comb space In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb spa ...
*
Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither L ...
*
Rose (topology) In mathematics, a rose (also known as a bouquet of ''n'' circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topo ...


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)