Carcassonne Completed City
A city in Carcassonne is completed when their are no places you can possibly add any additional tiles to grow that city.    

Three Cities, A, B, &amp; C.
Three Tiles that can be laid down (1,2, &amp; 3) in the Green Square..

 

So in this first picture; only city "B" is completed (3 tiles); as you cannot add any more tiles to that specific city; city A &amp; C are incomplete as you can put at least one more tile on both of those cities (Green Square).





In this picture you will see if we use Tile #1 in the green square; we are actually merging Cities A &amp; C; making a single city of 7 tiles.   City B is unchanged and is still considered completed.





In this picture we played tile #2, it completed both City A (4 tiles) and City C (4 tiles).  None of these cities can grow anymore so they are ALL considered completed.





Using tile #3, will actually keep both City A &amp; C open as they both have at least ONE place to add an additional tile.



Also depending on what additional add-ons you have added to your base game, you could also play a tile like 

 

that closes city C and leave city A open.