Abstract
Assuming that there exists in the unit interval [0, 1] a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff spaceK with the following properties: (i) For each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C(K), norm) is a denseG δ subset ofB, and (ii)C(K) does not admit a Kadeč norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.
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Namioka, I., Pol, R. Mappings of Baire spaces into function spaces and Kadeč renorming. Israel J. Math. 78, 1–20 (1992). https://doi.org/10.1007/BF02801567
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DOI: https://doi.org/10.1007/BF02801567