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’s fairly obvious that American education is a cultural flop. Americans are not a well-educated people culturally, and their vocational education often has to be learned all over again after they leave school and college. On the other hand, they have open quick minds and if their education has little sharp positive value, it has not the stultifying effects of a more rigid training.”
—Raymond Chandler (1888–1959)
“Since time immemorial, one the dry earth, scraped to the bone, of this immeasurable country, a few men travelled ceaselessly, they owned nothing, but they served no one, free and wretched lords in a strange kingdom. Janine did not know why this idea filled her with a sadness so soft and so vast that she closed her eyes. She only knew that this kingdom, which had always been promised to her would never be her, never again, except at this moment.”
—Albert Camus 1013–1960, French-Algerian novelist, dramatist, philosopher. Janine in Algeria, in The Fall, p. 27, Gallimard (9157)