Answer to Question #217674 in Differential Geometry | Topology for Prathibha Rose

Question

Prove that the fixed point property is a topological invariant

Answer 1

Solution:

Proof:

Let X be a topological space with the fixed-point property (FPP). Let Y be a topological space that is homeomorphic to X, so that there exists a homeomorphism g : X $\rightarrow$ Y . Let f : Y $\rightarrow$Y be an arbitrary continuous function. We want to show that f must have a fixed point, and thus Y has the FPP. Since g is a homeomorphism, it is continuous, and g^-1 must exist and also be continuous. Consider the function h = g^-1 o f o g : X $\rightarrow$ X. Since composition of continuous functions is continuous, h is a continuous function from X to X, and thus since X has the FPP, h has a fixed point.