$K$ and $L$ homeomorphic, then $C(K)$ is isomorphic to $C(L)$

Question

Can someone sketch the proof (or give me some reference) of the following fact : If $K$ and $L$, compact and Hausdorff spaces, are homeomorphic then the lattices $C(K)$ and $C(L)$ are isomorphic. (I am aware this is half of Kaplansky theorem, but I'm curious to know why)

Thank you.

Answer 1

Let $\xi :K\longrightarrow L$ be a homeomorphism. Then $T_{\xi }: C(L)\longrightarrow C(K),$ $T_{\xi } (u) =u\circ \xi $ is an isomorphism.