# Space with perfect fundamental group

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: perfect group*

*This property of topological spaces is defined as the property of the following associated group: first homology group having the following group property: trivial group*

## Definition

A topological space is said to have **perfect fundamental group** if it satisfies the following equivalent conditions:

- It is path-connected, and its fundamental group is perfect
- It is path-connected, and the first homology group is trivial

## Relation with other properties

### Stronger properties

Categories:

- Homotopy-invariant properties of topological spaces
- Properties of topological spaces
- Properties of topological spaces determined by fundamental group
- Properties of topological spaces for an associated group being a perfect group
- Properties of topological spaces determined by first homology group
- Properties of topological spaces for an associated group being a trivial group