# Classifying space

From Groupprops

## Definition

### Classifying space of an abstract group

A **classifying space** of a group is a topological space whose fundamental group is and whose universal covering space is weakly contractible.

Every group possesses a classifying space which is also a CW-space, viz., can be given a CW-complex structure, and in this case the universal covering space exists and is actually contractible.