Question about workout clothes and indoor shoes for Seoul gym

Hello! I'm going to Seoul next week, and I heard that the weather is still cold these days. So I'm wondering what clothes to bring when working out at the gym. I'm curious if the gym in Seoul is well heated. If the heating is not good, should I prepare long pants and long sleeves instead of shorts and sleeveless shirts? Also, I heard that you must bring indoor shoes when going to the gym in Seoul, and I would like to know if this rule applies to daily users as well. It's a small concern to pack light, but I think your experience could be a big help. I would appreciate your help!