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

Weaker properties

• 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