Topping lifts are lines that extend from the top of the mast to the end of the boom on sailboats. The lift is there to support the boom during certain points of sailing and when taking in the sail. Most boats don’t have a topping lift, why I don’t know. All I know is, not having one on most non-race boats is stupid. You need one to take in the sails. Having the sail lowered and the boom sweeping all over the deck and overboard makes you look like an idiot.
On small boats and two-masted boats of fairly light displacement, a simple single part lift is all you need. The lift can be made fast to the end of the boom and led aloft through a block and down the mast to a convenient cleat, or it can be attached to the mast and be cleated on the boom. The single part setup that I just described works fine for small boats unless the boom is unusually large.
Larger boats with larger heavier booms might need a 2-part purchase either forward (this seems to be very common in Europe) or at the after end of the boom. Large sloops and fishing schooners on the east coast of the US mostly used a single lift with many as many parts of the tackle arranged on the boom end. These boats however had large crews that knew what they were doing so I guess that this setup isn’t the best for boats that have small crews.
Coasters on the other hand were intended to run shorthanded; they often sailed with just a captain and mate. Coasters used double lifts, port and starboard, often in conjunction with lazyjacks. The purchases that give the mechanical advantage was hung underneath the spreaders aloft with the hauling parts leading to the deck. This was about the standard except for the West Coast lumber trade.
West Coast boats use a setup that combined the coaster and fishing boat rigs. They used a simple compound purchase at the boom with the hauling part led aloft and back down to the deck near the base of the mast. These lifts were often double sided with preventers and usually without lazyjacks. This setup is my favorite one, and on the boats that I’ve designed using this setup, the owners and crew seem very pleased with level of control and lack of windage aloft.
This is by no means a complete list. There seems to be as many different rig setups as there are fish in the sea. I’ve added these styles because they seemed to be very widespread at the time. Most of the boats from the great days of sail that I have been privileged to set foot on, used one of these topping lift systems.
I was referring to to the topping lift on mainsails. You do make a good point however. Personally my boat doesn't have a topping lift on a mainsail, but the chute
on it does.