Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is connected since it is the quotient of the upper half plane by the action of a subgroup (called $\Gamma_1(N)$) of $SL(2, \mathbb{Z})$.