Weakly contractible space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of contractibility. View other variations of contractibility

Definition

A topological space is said to be weakly contractible if all its homotopy groups are trivial. In other words, any map from a sphere to the given topological space is nullhomotopic.

Relation with other properties

Stronger properties

• Contractible space: The converse implication holds for CW-spaces, via Whitehead's theorem

Metaproperties

Products

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

Since the homotopy group of the product of two spaces is the product of their homotopy groups, the product of two weakly contractible spaces is again weakly contractible.

Retract-hereditariness

This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces

Any retract, and more generally, any homotopically injective subspace of a weakly contractible space is again weakly contractible.