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We prove that if a compact set E E in C N \mathbb {C}^N is contained in an arc J J , then there is a choice of J J whose polynomial hull J ^ \widehat {J} is J ∪ E ^ J\cup \widehat {E} . This strengthens an earlier result of the author. We also correct an inaccuracy in the statement, and fill a gap in the proof, of that earlier result.more » « less
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It is known that for $$X$$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ contains a dense $$G_\delta$$ set in the space $$C_b(X)$$ of all bounded continuous real-valued functions on $$X$$ in the supremum norm. Furthermore, when $$X$$ is separable, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $$X$$ is itself a $$G_\delta$$ set. We show that in contrast, when $$X$$ is nonseparable, this set of functions is not even a Borel set.more » « less
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null (Ed.)Abstract We strengthen, in various directions, the theorem of Garnett that every $$\unicode[STIX]{x1D70E}$$ -compact, completely regular space $$X$$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $$X$$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $$X$$ of a Euclidean space, there is a compact set $$K$$ in some $$\mathbb{C}^{N}$$ so that $$\widehat{K}\backslash K$$ contains a Gleason part homeomorphic to $$X$$ , and $$\widehat{K}$$ contains no analytic discs.more » « less
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