Theorem 5.27.3 (Gleason, 1958).label Let $X$ be a compact Hausdorff space, then the following are equivalent:

1. (1)

   $X$ is extremally disconnected.

2. (P)

   For any compact Hausdorff spaces $Y, Z$, surjective continuous map $\pi \in C(Y; Z)$, and $f \in C(X; Z)$, there exists a lifting $\tilde f \in C(X; Y)$ such that the following diagram commutes:

   \[\xymatrix{ & Y \ar@{->}[d]^{\pi} \\ X \ar@{->}[ru]^{\tilde f} \ar@{->}[r]^{f} & Z }\]

   
   
3. (I)

   For every pair of operator systems $S \subset T$ and completely positive map $\phi: S \to C(X; \complex)$, there exists a completely positive map $\Phi: T \to C(X; \complex)$ such that the following diagram commutes:

   \[\xymatrix{ T \ar@{->}[r]^{\Phi} & C(X; \complex) \\ S \ar@{^{(}->}[u] \ar@{->}[ru]_{\phi} & }\]