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Sums, products and continuity of Borel maps in nonseparable metric spaces


Author: R. W. Hansell
Journal: Proc. Amer. Math. Soc. 104 (1988), 465-471
MSC: Primary 28A05; Secondary 28A35, 28C10, 54H05
DOI: https://doi.org/10.1090/S0002-9939-1988-0962814-3
MathSciNet review: 962814
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Abstract: We give a comparatively simple proof that the complex map $ (f,g):T \to X \times Y$ will be (Borel measurable) of class $ \alpha + \alpha $, whenever $ f$ and $ g$ are of class $ \alpha $, for not necessarily separable metric spaces $ T,X$ and $ Y$. The Borel measurability of other types of maps, such as the sum of two vector-valued maps, is easily deduced from this. A general version of this result, applicable to abstract measurable spaces, is also proven.

Our second principal result shows that, if $ f:T \times X \to Y$ is continuous in each variable separately, where $ X$ and $ Y$ are metrizable, then $ f$ has a $ \sigma $-locally finite function base of closed sets, and thus will be continuous at each point of the complement of some first category subset of $ T \times X$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0962814-3
Keywords: Borel measurable maps, nonseparable metric spaces, complex maps, separately continuous maps
Article copyright: © Copyright 1988 American Mathematical Society

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