Contractibility is retract-hereditary

This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space) satisfying a topological space metaproperty (i.e., retract-hereditary property of topological spaces)
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Statement

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.

Definitions used

Contractible space

Further information: contractible space

Retract

Further information: Retract

Subspace topology

Further information: Subspace topology

Proof

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