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A note on weak-star and norm Borel sets in the dual of the space of continuous functions

Author: S. Ferrari
Journal: Proc. Amer. Math. Soc. 148 (2020), 2157-2161
MSC (2010): Primary 28A05, 54H05
Published electronically: January 29, 2020
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Abstract: Let $ \operatorname {Bo}(T,\tau )$ be the Borel $ \sigma $-algebra generated by the topology $ \tau $ on $ T$. In this paper we show that if $ K$ is a Hausdorff compact space, then every subset of $ K$ is a Borel set if and only if

$\displaystyle \operatorname {Bo}(C^*(K),\mathnormal {w}^*)=\operatorname {Bo}(C^*(K),\left \Vert\cdot \right \Vert),$

where $ \mathnormal {w}^*$ denotes the weak-star topology and $ \left \Vert{\cdot }\right \Vert$ is the dual norm with respect to the sup-norm on the space of real-valued continuous functions $ C(K)$. Furthermore, we study the topological properties of the Hausdorff compact spaces $ K$ such that every subset is a Borel set. In particular, we show that if the axiom of choice holds true, then $ K$ is scattered.

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Additional Information

S. Ferrari
Affiliation: Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

Keywords: Borel $\sigma$-algebra, weak-star topology, compact sets with only Borel subsets.
Received by editor(s): September 20, 2019
Received by editor(s) in revised form: October 11, 2019
Published electronically: January 29, 2020
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2020 American Mathematical Society