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:

    “When the inhabitants of some sequestered island first descry the “big canoe” of the European rolling through the blue waters towards their shores, they rush down to the beach in crowds, and with open arms stand ready to embrace the strangers. Fatal embrace! They fold to their bosoms the vipers whose sting is destined to poison all their joys; and the instinctive feeling of love within their breasts is soon converted into the bitterest hate.”
    —Herman Melville (1819–1891)

    “We are closed in, and the key is turned
    On our uncertainty;”
    —William Butler Yeats (1865–1939)