In topology, a branch of mathematics, a point x of a set S is called an isolated point of S if there exists a neighborhood of x not containing other points of S.
In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x which contains no other points of S.
Equivalently, a point x in S is an isolated point of S if and only if it is not a limit point of S.
A set which is made up only of isolated points is called a discrete set. Any discrete subset of Euclidean space is countable, since the isolation of each of its points (together with the fact the rationals are dense in the reals) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many. However, a set can be countable but not discrete, e.g. the rational numbers with the absolute difference metric. See also discrete space.
A set with no isolated point is said to be dense-in-itself. A closed set with no isolated point is called a perfect set.
The number of isolated points is a topological invariant, i.e. if two topological spaces and are homeomorphic, the number of isolated points in each is equal.
Read more about Isolated Point: Examples
Famous quotes containing the words isolated and/or point:
“The improved American highway system ... isolated the American-in-transit. On his speedway ... he had no contact with the towns which he by-passed. If he stopped for food or gas, he was served no local fare or local fuel, but had one of Howard Johnson’s nationally branded ice cream flavors, and so many gallons of Exxon. This vast ocean of superhighways was nearly as free of culture as the sea traversed by the Mayflower Pilgrims.”
—Daniel J. Boorstin (b. 1914)
“Mildred Pierce: You look down on me because I work for a living, don’t you? You always have. All right, I work. I cook food and sell it and make a profit on it, which, I might point out, you’re not too proud to share with me.
Monte Beragon: Yes, I take money from you, Mildred. But not enough to make me like kitchens or cooks. They smell of grease.
Mildred Pierce: I don’t notice you shrinking away from a fifty- dollar bill because it smells of grease.”
—Ranald MacDougall (1915–1973)