A note on weak-star and norm Borel sets in the dual of the space of continuous functions
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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 \[ \operatorname {Bo}(C^*(K),\mathnormal {w}^*)=\operatorname {Bo}(C^*(K),\left \|\cdot \right \|),\] where $\mathnormal {w}^*$ denotes the weak-star topology and $\left \|{\cdot }\right \|$ 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.References
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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
- MR Author ID: 1151838
- Email: simone.ferrari1@unipr.it
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 2157-2161
- MSC (2010): Primary 28A05, 54H05
- DOI: https://doi.org/10.1090/proc/14919
- MathSciNet review: 4078100