If $(X,\tau)$ has more than $1$ point and is $T\_2$ and connected, do we necessarily have $|X| =|\tau|$?



If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?
Consider the topology on $\mathbb{R}^2$ generated by subsets that are open in some line from the origin. This topological space is connected, has the cardinality of continuum, and has $2^c$ open subsets.


