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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