In Pokemon Red/Blue, we visit multiple cities and towns on our journey. Most of them have a single building with the GYM label, housing a Gym Leader. The only exception to this is Pallet Town, the players starting town; and Lavender Town, where it has the Pokemon Tower instead.
Correction
Um actually … Saffron City doesn’t have a single building with the GYM label. In addition to the official Psychic Gym, Saffron City has a fighting dojo, a building that also has the GYM label, even though it doesn’t have an official Gym status. Also, there is no Gym Leader inside, instead it is the Karate Master.
In Fire Red and Leaf Green, the dojo no longer has the GYM label, but it still has a GYM signage outside.
Ah right. I think my correction wasn’t clear. The point of contention is whether there is a single building with the GYM label. I’ve clarified the correction.