# Compact-by-contractible group

This article defines a property that can be evaluated for a topological group (usually, a T0 topological group)

View a complete list of such properties

## Definition

A topological group is termed **compact-by-contractible** if it has a closed compact subgroup such that the quotient space is contractible.

In the particular case where the quotient space is also paracompact Hausdorff, the compact subgroup is a topological factor, and the projection onto this factor is a strong deformation retraction. This follows from the fact that the vector bundle class functor is homotopy-invariant for paracompact Hausdorff spaces.