I think it also depends upon the cat. My little Kitten Mitten (RIP) was such a hyper little thing that the moment a doorway was opened, she had to be the first one to rush right on through; EVEN if it was a CLOSET door.
I can’t remember how many times this happened over and over every time I went to get something out of a closet. You think she might have gotten tired of rushing madly into closets, but nope. Every single time.
And getting out of my apt. was always a chore since she’d be right at my feet waiting to rush through the door. And since the only direction then to go was up, I had to follow her up several flights of stairs to bring her back home.
It finally dawned on me that she was quicker than I enough times to be annoying. So every time I left I would pick her up before going out, turn around and toss her back into the living room and shut the door before she could run back.
I guess she might not have lasted too long against predators out in the wild jungle with that lack of caution, but she was a friendly inquisitive little soul who immediately loved everyone she met.
She was really one of the most unique cats I ever had. Definitely different :)