Alexandroff-Hausdorff Theorem
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Problem IX.7.13 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IX.7 "Abelian von Neumann Algebras"
Show that every totally disconnected compact metric space is the continuous image of the Cantor ternary set. (Do this directly; do not try to use -algebras.) Combine this with Exercise 12 to get that every compact metric space is the continuous image of the Cantor set.
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Origin: Stated according to the original source
References
[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.