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

Proof details

Given: A topological space $X$ satisfying the fixed-point property, a retraction $r:X\to A$ where $A\subset X$ and $r(a)=a$ for all $a\in A$

To prove: $A$ satisfies the fixed-point property

Proof: Let $i$ denote the inclusion of $A$ in $X$.

Consider any continuous map $f:A\to A$. We need to show that $f$ has a fixed point in $A$. Consider the composition $g=i\circ f\circ r$. This is a map from $X$ to $X$ that first retracts to $A$, then applies $f$, and then views the resulting point of $A$ as a point in $X$. $g$ is a composite of continuous maps, so $g$ is continuous. Since $X$ has the fixed-point property, there exists $x\in X$ such that $g(x)=x$.

But by construction, $g(x)$ is actually inside $A$, so in fact $x\in A$. But if $x\in A$, $r(x)=x$, so we conclude that $x=g(x)=f(r(x))=f(x)$. Thus, $x\in A$ is a fixed point of $f$, completing the proof.