Contractibility is retract-hereditary

Property-theoretic statement
The property of topological spaces of being contractible is a retract-hereditary property of topological spaces.

Verbal statement
Any retract of a contractible space is contractible.

Proof outline

 * Consider a contracting homotopy for the whole space
 * Compose this with the retraction, and show that the composite is a contracting homotopy for the retract