If $(X,\tau)$ is a connected $T\_2$-space with $|X|=\aleph\_0$, what values can $|\tau|$ take?



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


