Simply connected 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

This property of topological spaces is defined as the property of the following associated group: fundamental group having the following group property: trivial group

Definition

Symbol-free definition

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

• It is path-connected, and any loop at any point is homotopic to the constant loop at that point
• It is path-connected, and its fundamental group is trivial

Definition with symbols

Fill this in later

Relation with other properties

Stronger properties

• Contractible space
• Weakly contractible space
• Multiply connected space

Weaker properties

• Semilocally simply connected space
• Simple space
• Space with Abelian fundamental group
• Space with perfect fundamental group