Does Chicago building code allow open ceiling joists in a finished basement?
Most 1 and 2 family units are non-rated construction. Triplexes (3 living units) and up require fire sprinklers, fire party walls, etc. (See ICC Chapter 3)

There are exceptions where the structure is built within the “setbacks” and the walls and portions of the floor, ceilings, roofs require fire rated construction. 

In your case, due to the proximity of the 3 units to each other, I’d guess you’d need fire walls (and doors, openings, etc.) between units, but not ceilings. 

If you have a Building Permit, I’d call your Building Department and give them your Permit number and ask if ceilings are required. They are there to help and don’t want a redo either.