Cardinality of closed discrete subspace of a Tychonoff space with Countable Chain condition

Question

If $X$ is a Tychonoff space satisfying countable chain condition, then is it possible that $X$ has an uncountable closed and discrete subspace.

Answer 1

Yes. The Sorgenfrey plane is an example: it is ccc since it is separable, and the antidiagonal $\{(x,-x):x\in\mathbb{R}\}$ is an uncountable closed discrete subspace.