Nowhere Dense Set - Open and Closed

Open and Closed

  • A nowhere dense set need not be closed (for instance, the set is nowhere dense in the reals), but is properly contained in a nowhere dense closed set, namely its closure (which would add 0 to the set). Indeed, a set is nowhere dense if and only if its closure is nowhere dense.
  • The complement of a closed nowhere dense set is a dense open set, and thus the complement of a nowhere dense set is a set with dense interior.
  • The boundary of every open set is closed and nowhere dense.
  • Every closed nowhere dense set is the boundary of an open set.

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