Forum:Costs and Priority

If a situation occurs where a player summons a monster with a cost effect, and the opposing player activates a card to destroy the monster, can he/she retain priority, even though it is a cost? (priority rulings http://yugioh.wikia.com/wiki/Priority)

Player A summons "the big saturn" (http://yugioh.wikia.com/wiki/The_Big_Saturn) Player B activates Bottomless trap hole Player A states that he wishes to retain priority and activate the effect.