Properties of Closed Sets
A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than 2.
Any intersection of closed sets is closed (including intersections of infinitely many closed sets), and any union of finitely many closed sets is closed. In particular, the empty set and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.
Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed.
Read more about this topic: Closed Set
Famous quotes containing the words properties of, properties, closed and/or sets:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“We are closed in, and the key is turned
On our uncertainty;”
—William Butler Yeats (1865–1939)
“The believing mind reaches its perihelion in the so-called Liberals. They believe in each and every quack who sets up his booth in the fairgrounds, including the Communists. The Communists have some talents too, but they always fall short of believing in the Liberals.”
—H.L. (Henry Lewis)