Are the irrationals Gδ because they are uncountable?

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https://math.stackexchange.com/a/61110/1248511 I just saw this proof for showing that the rationals are not a Gδ set(without using baire's theorem) and while I'm not really sure about what exactly I'm asking, I wanted to know if there's any relation between the uncountability of the irrationals and the fact that they can be expressed as the intersection of open sets