Cardinality of the topology in countable connected $T_2$-spaces

Question

If $(X,\tau)$ is a connected $T_2$-space with $|X|=\aleph_0$, what values can $|\tau|$ take?

Answer 1

Continuum. Connected or not, every $T_2$ space has an infinite pairwise disjoint family of (non-empty) open sets. All unions of all possible subfamilies will give you $\mathfrak{c}$ many open sets (and more is not possible in a countable space).