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
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.