18 - The theorems of Brouwer and Schauder

Summary

We have been using the term ‘fixed point property’ (f.p.p.) as it relates to the class of nonexpansive mappings but, of course, this property may be applied to any class of mappings. The ‘topological’ fixed point property is of fundamental importance in the broader context of fixed point theory.

Definition 18.1 A topological space X is said to have the (topological) fixed point property (t.f.p.p. or f.p.p. if the meaning is clear) if each continuous mapping T: X→X has a fixed point.

It is not surprising that the above property is a topological invariant.

Lemma 18.1 If X and Y are homeomorphic and if X has the t.f.p.p., then Y also has the t.f.p.p.

Proof Let h: X→Y be a homeomorphism with h(X) = Y, and suppose f: Y→Y is continuous. Then g = h^–1 ∘ f ∘ h: X→X and g is continuous; hence there exists x ∈ X such that gx = x, implying y = hx = fy.

Another very useful observation about the t.f.p.p. is the following:

Lemma 18.2 If a topological space X has the t.f.p.p. and if Y is a retract of X, then Y has the t.f.p.p.

Proof. Let r be a retraction of X onto Y and let f: Y→Y be continuous. Then f may be extended to a continuous mapping g = f ∘ r: X→X. Any fixed point of g must also be a fixed point of f.