In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X is contractible if the identity map

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

 on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point.

A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

s of a contractible space are trivial

Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology

Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....

 is a homotopy invariant, the reduced homology groups

Reduced homology

In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero...

 of a contractible space are all trivial.

For a topological space X the following are all equivalent (here Y is an arbitrary topological space):

• X is contractible (i.e. the identity map is null-homotopic).
• X is homotopy equivalent to a one-point space.
• X deformation retract
  Deformation retract
  In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...
  
  s onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)
• Any two maps f,g : Y → X are homotopic.
• Any map f : Y → X is null-homotopic.

The cone

Cone (topology)

In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:CX = /\,of the product of X with the unit interval I = [0, 1]....

 on a space X is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).

Furthermore, X is contractible if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 there exists a retraction

Deformation retract

In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...

 from the cone of X to X.

Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected

N-connected

In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".-n-connected space:...

 for all n ≥ 0.

Locally contractible spaces

A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not necessarily locally contractible nor vice-versa. For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected

Locally connected space

In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...

, and locally connected.

Examples and counterexamples

• Any star domain
  Star domain
  In mathematics, a set S in the Euclidean space Rn is called a star domain if there exists x_0 in S such that for all x in S the line segment from x_0 to x is in S...
  
   of a Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   is contractible.
• The Whitehead manifold
  Whitehead manifold
  In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture....
  
   is contractible.
• Spheres of any finite dimension are not contractible.
• The unit sphere
  Unit sphere
  In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
  
   in an infinite-dimensional Hilbert space
  Hilbert space
  The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
  
   is contractible.
• The house with two rooms
  House with two rooms
  House with two rooms or Bing's house is a particular contractible 2-complex that is not collapsible.The name was given by R. H. Bing.-External links:*...
  
   is standard example of a space which is contractible, but not intuitively so.
• Dunce hat
  Dunce hat (topology)
  For the item of clothing designed to be humiliating, now rarely used, see dunce cap.In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed...
  
  
• The cone on a Hawaiian earring
  Hawaiian earring
  In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane R2 with center and radius 1/n for n = 1, 2, 3, ......
  
   is contractible (since it is a cone), but not locally contractible or even locally simply connected.

• All manifold
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
  s and CW complex
  CW complex
  In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
  
  es are locally contractible, but in general not contractible.