Limit point

In mathematics, a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.

Definition

Let S be a subset of a topological space X. A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself. Note that it doesn't make a difference if we restrict the condition to open neighbourhoods only.

This is equivalent, in a T1 space, to requiring that every neighbourhood of x contains infinitely many points of S. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.

Alternatively, if the space X is Fréchet-Urysohn, we may say that x X is a limit point of S if and only if there is an ω-sequence of points in S \ {x} whose limit is x; hence, x is called a limit point.

Types of limit points

A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.
With respect to the usual Euclidean topology, the sequence of rational numbers xn = (-1)n·n/n+1 has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1.

If every open set containing x contains infinitely many points of S then x is a specific type of limit point called an ω-accumulation point of S.

If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.

If every open set U containing x satisfies |U S| = |S| then x is a specific type of limit point called a complete accumulation point of S.

A point x X is a cluster point or accumulation point of a sequence (xn)n  N if, for every neighbourhood V of x, there are infinitely many natural numbers n such that xn  V. If the space is Fréchet–Urysohn, this is equivalent to the assertion that x is a limit of some subsequence of the sequence (xn)n  N. The set of all cluster points of a sequence is sometimes called a limit set.

The concept of a net generalizes the idea of a sequence. Let be a net, where is a directed set. The point is said to be a cluster point of the net if for any neighbourhood of and any , there is some such that , equivalently, if has a subnet which converges to . Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for the related topic of filters.

Some facts

References

External links

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