( Separable ∧ $\omega$C ) ⇒ Countable

The space $X$ contains a countable dense subset, which is closed by $\omega$C, hence equal to $X$. So the space is countable.

• Converse
• References

The converse ( Countable ⇒ ( Separable ∧ $\omega$C )) cannot be proven or disproven from other known theorems