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.

Read more about this topic:  Nowhere Dense Set

Famous quotes containing the words open and/or closed:

    “Feminism, like Boston, is a state of mind. It is the state of mind of women who realize that their whole position in the social order is antiquated, as a woman cooking over an open fire with heavy iron pots would know that her entire housekeeping was out of date.”
    —Rheta Childe Dorr (1866–1948)

    “On a flat road runs the well-trained runner,
    He is lean and sinewy with muscular legs,
    He is thinly clothed, he leans forward as he runs,
    With lightly closed fists and arms partially raised.”
    —Walt Whitman (1819–1892)