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It is known that closed sets of $\mathbb{R}$ satisfies continuum hypothesis, that is, every closed subset of $\mathbb{R}$ is either countable or of the cardinality of the continuum.

Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?

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Yes. In fact the cardinality of every Borel set is countable or size continuum.

There is an easy proof for $G_\delta$. Every $G_\delta$ set is completely metrizable, and therefore either countable or contains a perfect set (which has a copy of a Cantor space).

(See this answer for the former fact, and the Cantor-Bendixson derivative, and related theorems for the latter)

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