Closed, I used to keep it open all the time when I was younger though. It's funny you should mention the fear thing, 'cause I've always found an open door to be more creepy than a closed one due to the thought of someone being able to creep in more silently.
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I'm never really concerned about someone creeping into my room without me being able to detect them. Even when i'm going to sleep, like I said, I keep one eye open, and listen to everything around me as closely as I can. That's reasonable, though.