Simply connected space

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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

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Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Contractible space homotopy-equivalent to a point contractible implies simply connected simply connected not implies contractible |
Weakly contractible space all homotopy groups trivial weakly contractible implies simply connected simply connected not implies weakly contractible |
Multiply connected space first few homotopy groups trivial

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Semilocally simply connected space
Simple space |
Space with abelian fundamental group
Space with perfect fundamental group

Metaproperties

Products

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

An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups.

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

A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space.