TITLE: A reference to a well-known characterization of scattered compact spaces
QUESTION [8 upvotes]: It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$.
Surprisingly, but I cannot find a good reference to this well-known fact (desirably some textbook). 
In the survey paper "Scattered spaces" in Encyclopedia of General Topology this fact is not mentioned, unfortunately.

REPLY [13 votes]: The proof in the direction that there is no continuous surjection from a compact scattered $X$ onto $[0,1]$ may be found here (Theorem 1):
W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42.
And the full characterisation may be found in (Theorem 8.5.4, p. 148):
Z. Semadeni, Banach spaces of continuous functions, vol. 1, PWN - Polish Scientific Publishers, Warsaw 1971.