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.
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Famous quotes containing the words isolated and/or point:
“Literature is not exhaustible, for the sufficient and simple reason that a single book is not. A book is not an isolated entity: it is a narration, an axis of innumerable narrations. One literature differs from another, either before or after it, not so much because of the text as for the manner in which it is read.”
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“The realization that he is white in a black country, and respected for it, is the turning point in the expatriate’s career. He can either forget it, or capitalize on it. Most choose the latter.”
—Paul Theroux (b. 1941)