Headspace, also known as the Dream World, is the imaginary world that OMORI and his friends reside in.
Headspace is largely assumed to be an imaginary world that Sunny created after being traumatized by Mari's death. However, there are some clues throughout the game that imply Headspace could be more than Sunny's imagination. Mr. Outback states,
"Well, this place has been around since the beginning of time, I reckon. It looks different to everyone… but this time around, it’s pretty colorful and cutesy for the most part… Not to mention pretty dang big! That’s a powerful imagination you’ve got there, kiddo."
Black Space is potentially a part of Headspace, as there are many entrances into sections of Black Space throughout Headspace.