Contractible space

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

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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

Definition with symbols

A topological space $X$ is said to be contractible if there is 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

Weaker properties

• Simply 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