Is there a ZFC example of a Tychonoff space $X$ such that:

- $X$ is separable.
- $X$ has countable tightness (that is, a subset of $X$ is closed if and only if it contains the closure of each one of its countable subsets)
- $X$ has cardinality larger than the continuum?

The existence of a separable space of countable tightness and cardinality larger than the continuum is consistent with ZFC. Take any hereditarily separable space of cardinality larger than the continuum, like Fedorchuk's compact $S$-space. Hereditary separability implies both separability and countable tightness.

However, an example answering my question cannot be hereditarily separable. By Todorcevic's Theorem, every hereditarily separable regular space is hereditarily Lindelof under PFA and every hereditarily Lindelof space has cardinality bounded by the continuum.

Note also that an example answering my question cannot be compact. In fact, by Balogh's Theorem, every compact Hausdorff space of countable tightness is sequential under PFA, and it is easy to see that every separable sequential space has cardinality bounded by the continuum.

Finally note that to find a Hausdorff non-regular counterexample it suffices to take the Katětov extension of the integers. Let $U(\omega)$ be the set of all non-principal ultrafilters on $\omega$ and define a topology on $X=\omega \cup U(\omega)$ by declaring every point of $\omega$ to be isolated and a basic neighbourhood of a point $p \in U(\omega)$ to be $\{p\} \cup A$, where $A \in p$. Then $X$ is separable, has countable tightness and $|X|=2^\mathfrak{c}$.