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Let $(X,\tau)$ be a separable topological space.

What can we conclude about the cardinality of $X$?

$X$ must be finite.

$X$ must be countable.

$X$ must have cardinality at most that of the continuum, i.e., $X$ must be in bijective correspondence with $\mathbb{R}$.

The cardinality of $X$ is at most that of $\mathbb{R}^\mathbb{R}$, the set of all functions from $\mathbb{R}$ to itself.

Nothing. $X$ may have any cardinality.