Contractible space

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces

Definition

Symbol-free definition

A topological space is said to be contractible if it satisfies the following equivalent conditions:

• It is in the same homotopy class as a point
• The identity map from the space to itself, is homotopic to a constant map, from the space to a particular point in that space
• There is a single point which is a homotopy retract
• It has a contracting homotopy

Definition with symbols

A topological space $X$ is said to be contractible if it has a contracting homotopy, viz a continuous map $f:X\times [0,1]\to X$ and a point $x_{0}\in X$ such that $f(x,0)=x$ and $f(x,1)=x_{0}$ for all $x$.

Relation with other properties

Stronger properties

• Cone space over some topological space For full proof, refer: Cone space implies contractible
• Suddenly contractible space
• SDR-contractible space

Weaker properties

• Weakly contractible space
• Multiply connected space
• Simply connected space
• Path-connected space
• Connected space

Metaproperties

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

A direct product of contractible spaces is contractible. For full proof, refer: Contractibility is direct product-closed

Subset-related properties

Closure

The closure of a contractible subset need not be contractible; in fact it need not even be path-connected. An example is the topologist's sine curve, which is contractible as it is the image of a path; however, it's closure is not even path-connected.

Interior

The interior of a contractible subset need not be contractible; in fact, it need not even be connected. An example is a wedge of two discs, whose interior is the disjoint union of their interiors.

Intersection

An intersection of contractible subsets need not be contractible. There is, however, a relation between the homology of the intersection and the homology of the union, when the contractible subsets are open (or more generally, when they are strong deformation retracts of neighbourhoods).

An example is the intersection of two semicircular paths on the circle, which is a pair of points.

Connected union

A connected union of contractible subsets need not be contractible.