Actually, I have tried the obvious fact that a subset of the line is connected iff it's an interval. And, the family of all possible intervals of the line is equinumerous with $\Bbb R$, as we can send any interval to an ordered pair having order as that of the end-points , including both the infinities.
But, for the plane, I am guessing about a suitable characteristic to match any connected set except the general definition. I thought the compliment, the cardinality of the family of sets which can be written as a separation, both non-empty, but then I think I need to find cardinality of all open subsets of the plane .
But, I need a hint to approach concretely .