ORDER PRESERVING BIJECTIONS OF C+(X)

Abstract

Let $X$ be a compact Hausdorff space which satisfies the first axiom of countability, and $C_+(X)$ the set of all continuous functions from $X$ to $[0,\infty)$. If $\varphi : C_{+}(X) \rightarrow C_{+}(X)$ is a bijective map which preserves the order in both directions, then there exists a homeomorphism $\omega : X \mathcal{\rightarrow } X$ and for each $x \in X$ a bijective, increasing map $m_{x} : [0,\infty) \mathcal{\rightarrow } [0,\infty)$ such that $\varphi(f)(x) = m_{x}(f(\omega(x)))$, for all $x \in X$ and $f \in C_{+}(X)$.