Fixed-point property is retract-hereditary

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This article gives the statement, and possibly proof, of a topological space property (i.e., fixed-point property) 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 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.

Definitions used

Fixed-point property

Retract

Subspace topology

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.