With top set on a monotone board, checking back allows the opponent to bluff or catch up, as we block their strongest calling range.
Checking is the preferred play here. While we have top set, we heavily block the Ace-high hands SB would use to continue, and the monotone texture makes betting less effective for value.
Note: Betting is fine, but checking back top set on a monotone board is higher EV as it protects our range and lets SB bluff into us.
After being called on the flop, checking back the turn is a strong option to control the pot and induce river bluffs on a board where the flush is already possible.
Note: Betting is slightly lower EV than checking; we want to keep SB's range wide and avoid bloating the pot against completed flushes.
Once SB raises, we are in a high-variance spot. Shoving is a viable mix to maximize value from worse sets or two pairs that might overplay, though calling is also reasonable to keep their bluffs in.