The T terminal is a 5V current limited supply. The controller measures the current on the T terminal, and that is what controls compressor speed and on/off. It should be described in one of my prior posts in detail.
Connecting the T terminal to ground or C sets the current to max, which is around 5mA. This is the lowest speed I believe, and any current between 5 and 2ma varies the speed. When the current drops too low, (such as disconnecting the T wire) the compressor should stop. Most mechanical thermostats for these units have a series resistor, and they close to turn the unit on, and open to turn it off.
If you have some type of fault, you can connect an LED between two pins on the controller (I don't recall which, read the manual). It will then blink a code, which will indicate the fault.