We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K,X) is isomorphic to C(𝓒,X) where 𝓒 denotes the Cantor set. For X = ℝ, this gives the well known Milyutin Theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-1,
author = {Alexander D. Arvanitakis},
title = {A simultaneous selection theorem},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {1-14},
zbl = {1257.54025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-1}
}
Alexander D. Arvanitakis. A simultaneous selection theorem. Fundamenta Mathematicae, Tome 219 (2012) pp. 1-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-1/