Fixed-point property is retract-hereditary

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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Statement

Property-theoretic statement

The property of topological spaces called the fixed-point property is a retract-hereditary property of topological spaces.

Verbal statement

Any retract of a topological space having the fixed-point property, also has the fixed-point property.

Proof

Proof outline

• Consider a self-map of the retract
• Compose with the retraction to get a self-map of the whole space
• Find a fixed point, and observe that it must be a fixed point of the original self-map