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.
Read more about this topic: Nowhere Dense Set
Famous quotes containing the words open and/or closed:
“It is open to a war resister to judge between the combatants and wish success to the one who has justice on his side. By so judging he is more likely to bring peace between the two than by remaining a mere spectator.”
—Mohandas K. Gandhi (1869–1948)
“Pray but one prayer for me ‘twixt thy closed lips,
Think but one thought of me up in the stars.”
—William Morris (1834–1896)