What happens when a MOSFETs gate is left open?
Not even close. The complete opposite, in fact.

A high-impedance floating N-channel gate can pick up charge from a variety of sources, (via the parasitic gate-to-drain capacitance, via handling - touching the gate with your finger, etc.) and spontaneously turn itself on. Any gate that could have a high-impedance control (a digital I/O from a micro, for instance) really needs a gate-to-source resistor to bleed off these spurious charges and keep the device state defined (=off) when the control is high-Z.