Abstract
We study Borel measurability of the spectrum in topological algebras. We give some equivalences of the various properties, show that the spectrum in a Banach algebra is continuous on a dense G δ {G_\delta } , and prove that in a Polish algebra the set of invertible elements is an F σ δ {F_{\sigma \delta }} and the inverse mapping is a Borel function of the second class.
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