# Simply connected space

From Topospaces

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

## Contents

## 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

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|FULL LIST, MORE INFO |

Weakly contractible space | all homotopy groups trivial | weakly contractible implies simply connected | simply connected not implies weakly contractible | |FULL LIST, MORE INFO |

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 | |FULL LIST, MORE INFO | |||

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.