Let $X$ be a compact space and $f:X\to X$ Be continuous,then $f:P(X)\to P(X)$$(A\mapsto f(A))$ has FP.($f(A)=A,A$ is no empty).

Let $X$ be a nonempty compact Hausdorff space, and $f\colon X\to X$ be continuous. Denote by $\mathcal P(X)$ the powerset of $X$. Then the function $f^+\colon\mathcal P(X)\to\mathcal P(X)$ defined by $f^+(A)=f[A]$ has a fixed point $f^+(A)=A$, where $A\subseteq X$ is nonempty and closed.