"My neighbor says you have to leave the oven door open when you broil. Is that true?"
A general rule of thumb is that you leave the oven door open when broiling in an electric oven, but closed to broil in a gas oven.
Here's why: A gas oven broils at a higher temperature, which can cause more smoke. Leave the oven door open, and that smoke can end up in your kitchen. Close it, and smoke escapes through the vent. Importantly, gas broilers will maintain a steady temperature with the oven door open or closed. An electric ovens, on the other hand, needs the cooler air in the kitchen to keep it from shutting down when it gets too hot. If you close the door when broiling in an electric oven, the temperature will cycle, making an uneven cooking process.
For recipe ideas, see 15-Minute Meals from the Broiler.