BOYFRIEND OR NOT??

so someone who i work with has just recently got out of a relationship and we were close anyway but just as friends. now it’s took a different turn and we both like eachother. he’s stayed over a couple times and we go out quite abit and we basically act like a couple. he also calls me babe etc and calls me his better half. his past relationship was awful, she cheated on him multiple times and used to say horrible things to him all the time and in the end he had enough and ended things. (she also admitted o cheating) however i don’t know if we are together or not and i don’t know how to bring the conversation up without it sounding like i’m pushing. i just don’t know what he’s told people and don’t want to make him look bad by saying something different but i also don’t want to ask him and him feel like i’m pushing for a relationship because i’m not atm. i’m quite happy with the way we’re going and just taking things slowly but i’m just very confused rn. any advice would be much appreciated!!