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:
“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)
“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)
“Thus piteously Love closed what he begat:
The union of this ever-diverse pair!
These two were rapid falcons in a snare,
Condemned to do the flitting of the bat.”
—George Meredith (1828–1909)
“Almsgiving tends to perpetuate poverty; aid does away with it once and for all. Almsgiving leaves a man just where he was before. Aid restores him to society as an individual worthy of all respect and not as a man with a grievance. Almsgiving is the generosity of the rich; social aid levels up social inequalities. Charity separates the rich from the poor; aid raises the needy and sets him on the same level with the rich.”
—Eva Perón (1919–1952)