What Is the Fixed Point Property in Topology?
Definition
For a function $f : X \to X$, an $x_{0}$ satisfying $f(x_{0}) = x_{0}$ is called a fixed point of $f$. If every continuous function $f$ has a fixed point, then $X$ is said to have the fixed point Property.
Explanation
It is mainly closely related to complete spaces.
At least in $\mathbb{R}$, using the intermediate value theorem, one can show that for $f : [a,b] \to [a,b]$ there always exists a $c$ satisfying $f(c) = c$.
Theorem
The fixed point property is a topological property.
Proof
Suppose there exists a homeomorphism $ h : X \to Y$ and that $X$ has the fixed point property. The proof is complete if we show that $Y$ has the fixed point property.
Let $f : Y \to Y$ be a continuous function, and define $g : X \to X$ by $g(x) = (h^{-1} \circ f \circ h) (x)$; then $g$ is also a continuous function. Since $X$ has the fixed point property, a fixed point $x_{0}$ of $g$ exists, and let $h(x_{0}) = y_{0} \in Y$. 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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