Extremally disconnected space Can one view $\ell_{\infty}^{4}$, $\mathbb{R}^{4}$ equipped with the $\ell_{\infty}$-norm, as a space of continuous functions
on some extremally disconnected space? and what would the extremally disconnected space be?
Thanks!
 A: Take the four point space $X = \{0,1,2,3\}$ with the discrete topology. Then $C(X,\mathbb{R}) = \ell^{\infty}(X,\mathbb{R})$ is the space $\mathbb{R}^4$ equipped with the $\sup$-norm. Obviously, $\{0,1,2,3\}$ is extremally disconnected, as every subset is closed and open.
Added: Of course, this generalizes to all finite dimensions. If $n = \{0,1,2,\ldots,n-1\}$ is the $n$-point space with the discrete topology, then $C(n) = \ell^{\infty}(n)$ is $\mathbb{R}^n$ with the $\sup$-norm.
There is no wiggle room here: by a theorem of Banach (metric case) Stone (general case) two spaces $C(K)$ and $C(L)$ with $K$ and $L$ compact Hausdorff are isometrically isomorphic if and only if $K$ and $L$ are homeomorphic. The proof is not hard: if $T: C(K) \to C(L)$ is an isometric isomorphism then its adjoint is a homeomorphism from the unit ball of $C(L)^{\ast}$ to the one of $C(K)^{\ast}$ in the weak$^{\ast}$-topology. Then one finishes off by observing that $K$ and $L$ are stored as extremal points of the unit balls. Details can be found e.g. in Theorem 2 of §3 in Chapter V on page 115 of Day's Normed Linear Spaces, 3rd edition, Springer, 1973.
