The local compactness follows from the descriptions of the idele group as a restricted product.
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Therefore local compactness characterizes finite-dimensionality.
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There are also more general versions of the Stone Weierstrass that weaken the assumption of local compactness.
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Notice how, to preserve properties such as local connectedness, second countability, local compactness etc & we require that the map be not only continuous but also open.
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The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets and was initially considered to be somewhat pathological due to the absence of a number of properties such as local compactness which are so frequently useful in analysis.
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