Limit point

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In mathematics, informally speaking, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x as well as one pleases. 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 adding its limit points.

A related concept is cluster point or accumulation point of a sequence.

1 Definition
2 Types of limit points
3 Cluster point
4 Some facts
5 See also
6 References

[edit] Definition

Let S be a subset of a topological space X. We say that a point x in X is a limit point of S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself. (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.)

[edit] Types of limit points

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

[edit] Cluster point

If X is a metric space with distance d, then a point x in X is a cluster point or accumulation point of a sequence (x_n ) if for every ε > 0, there are infinitely many values of n such that d (x,x_n ) < ε. Equivalently, that every neighborhood of x contains x_n for infinitely many n.

A limit point of the set of points in a sequence is a cluster point of the sequence. However, if for infinitely many n the values of x_n are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence.

A cluster point of a sequence is a subsequential limit: the limit of some subsequence.

The concept of a net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points:

If φ is a net on X based on directed set D and A is a subset of X, then φ is frequently in A if for every α in D there exists some β ≥ α, β in D, so that φ(β) is in A. A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.

Clustering and limit points are also defined for the related topic of filters.

The set of all cluster points of a sequence is sometimes called a limit set.

[edit] Some facts

We have the following characterisation of limit points: x is a limit point of S if and only if it is in the closure of S \ {x}.
- Proof: We use the fact that a point is in the closure of a set if and only if every neighbourhood of the point contains a point of the set. Now, x is a limit point of S, if and only if every neighbourhood of x contains a point of S other than x, if and only if every neighbourhood of x contains a point of S \ {x}, if and only if x is in the closure of S \ {x}.

If we use L(S) to denote the set of limit points of S, then we have the following characterisation of the closure of S: The closure of S is equal to the union of S and L(S).
- Proof: ("Left subset") Suppose x is in the closure of S. If x is in S, we are done. If x is not in S, then every neighbourhood of x contains a point of S, and this point cannot be x. In other words, x is a limit point of S and x is in L(S). ("Right subset") If x is in S, then every neighbourhood of x clearly meets S, so x is in the closure of S. If x is in L(S), then every neighbourhood of x contains a point of S (other than x), so x is again in the closure of S. This completes the proof.

A corollary of this result gives us a characterisation of closed sets: A set S is closed if and only if it contains all of its limit points.
- Proof: S is closed if and only if S is equal to its closure if and only if S = S ∪ L(S) if and only if L(S) is contained in S.
- Another proof: Let S be a closed set and x a limit point of S. Then x must be in S, for otherwise the complement of S would be an open neighborhood of x that does not intersect S. Conversely, assume S contains all its limit points. We shall show that the complement of S is an open set. Let x be a point in the complement of S. By assumption, x is not a limit point, and hence there exists an open neighborhood U of x that does not intersect S, and so U lies entirely in the complement of S. Hence the complement of S is open.

No isolated point is a limit point of any set.
- Proof: If x is an isolated point, then {x} is a neighbourhood of x that contains no points other than x.

A space X is discrete if and only if no subset of X has a limit point.
- Proof: If X is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if X is not discrete, then there is a singleton {x} that is not open. Hence, every open neighbourhood of {x} contains a point y ≠ x, and so x is a limit point of X.

If a space X has the trivial topology and S is a subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.
- Proof: As long as S \ {x} is nonempty, its closure will be X. It's only empty when S is empty or x is the unique element of S.

If a sequence in a space X converges to a point L in X and the elements are different from L then L is a limit point of X.

If L is a limit point of space X then there is not necessarily a sequence in X\{L} converging to L. For example, the smallest uncountable ordinal number omega-one (ω₁) is a limit point (with respect to the order topology) of the set of countable ordinal numbers, but a convergent sequence of countable sets has a limit that is not larger than the union of those sets, which is countable.