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arXiv · 1907.00679

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

S. Ferrari
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Abstract

Let $Bo(T,\tau)$ 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, $$Bo(C^*(K),w^*)=Bo(C^*(K),\|\cdot\|);$$ where $w^*$ denotes the weak-star topology and $\|\cdot\|$ 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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S. Ferrari. 2019-07-01. A note on weak-star and norm Borel sets in the dual of the space of continuous functions. https://doi.org/10.1090/proc/14919

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