# Space with abelian fundamental group

From Topospaces

(Redirected from Space with Abelian fundamental group)

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: Abelian group*

## Contents

## Definition

A topological space is said to have **Abelian fundamental group** if it is path-connected and the fundamental group at any point is Abelian.

## Relation with other properties

### Stronger properties

- Simply connected space
- Simple space
- Path-connected topological group

## Metaproperties

### Products

This property of topological spaces is closed under taking arbitrary products

View all properties of topological spaces closed under products

This corresponds to the algebraic fact that a direct product of Abelian groups is Abelian.

### 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

This corresponds to the algebraic fact that a group-theoretic retract of an Abelian group is Abelian.