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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.
Alexander D. Arvanitakis. "A simultaneous selection theorem." Fundamenta Mathematicae 219.1 (2012): 1-14. <http://eudml.org/doc/282856>.
@article{AlexanderD2012, abstract = {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.}, author = {Alexander D. Arvanitakis}, journal = {Fundamenta Mathematicae}, keywords = {selection theorem; extension theorem}, language = {eng}, number = {1}, pages = {1-14}, title = {A simultaneous selection theorem}, url = {http://eudml.org/doc/282856}, volume = {219}, year = {2012}, }
TY - JOUR AU - Alexander D. Arvanitakis TI - A simultaneous selection theorem JO - Fundamenta Mathematicae PY - 2012 VL - 219 IS - 1 SP - 1 EP - 14 AB - 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. LA - eng KW - selection theorem; extension theorem UR - http://eudml.org/doc/282856 ER -