📂Topology

What is the Fixed Point Property in Topology?

Definition

A fixed point of a function $f : X \to X$ is $x_{0}$ that satisfies $f(x_{0}) = x_{0}$ for $f$. It is said that $X$ has the fixed point property if every continuous function $f$ has a fixed point.

Explanation

It is closely related to complete spaces.

At least, in $\mathbb{R}$, it is possible to show that there always exists $c$ satisfying $f(c) = c$ for $f : [a,b] \to [a,b]$ using the intermediate value theorem.

Theorem

The fixed point property is a topological property.

Proof

Suppose that there is a homeomorphic mapping $h : X \to Y$ and $X$ has the fixed point property. To show that $Y$ has the fixed point property ends the proof.

If $f : Y \to Y$ is defined as a continuous function, and $g : X \to X$ is defined as $g(x) = (h^{-1} \circ f \circ h) (x)$, then $g$ is also a continuous function. Since $X$ has the fixed point property, there must exist a fixed point $x_{0}$ of $g$, let it be $h(x_{0}) = y_{0} \in Y$. Then $$\begin{align*} f (y_{0}) =& f( ( h (x_{0} ) ) \\ =& h \circ h^{-1} \circ f \circ h (x_{0}) \\ =& h(g(x_{0})) \\ =& h (x_{0}) \\ =& y_{0} \end{align*}$$, and $Y$ has the fixed point property.

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