# Limit point

In mathematics, a **limit point** (or **cluster point** or **accumulation point**) of a set *S* in a topological space *X* is a point *x* 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. A limit point of a set *S* does not itself 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.

There is also a closely related concept for sequences. A **cluster point** (or **accumulation point**) of a sequence (*x*_{n})_{n ∈ N} in a topological space *X* is a point *x* such that, for every neighbourhood *V* of *x*, there are infinitely many natural numbers *n* such that *x _{n}* ∈

*V*. This concept generalizes to nets and filters.

## Definition

Let *S* be a subset of a topological space *X*.
A point *x* in *X* is a **limit point** (or **cluster point** or **accumulation 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. 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.

If *X* is a *T*_{1} space (which all metric spaces are), then *x* ∈ *X* is a limit point of *S* if and only if every neighbourhood of *x* contains infinitely many points of *S*. Indeed, *T*_{1} spaces are characterized by this property.

If *X* is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then *x* ∈ *X* is a limit point of *S* if and only if there is a sequence of points in *S* \ {*x*} whose limit is *x*. Indeed, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of *S* is called a derived set of *S*.

## 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 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**.

## For sequences and nets

In a topological space , a point is said to be a **cluster point** (or **accumulation point**) of a sequence if, for every neighbourhood of , there are infinitely many such that . It is equivalent to say that for every neighbourhood of and every , there is some such that . If is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then is cluster point of if and only if is a limit of some subsequence of .
The set of all cluster points of a sequence is sometimes called the limit set.

The concept of a net generalizes the idea of a sequence. A net is a function , where is a directed set and is a topological space. A point is said to be a **cluster point** (or **accumulation point**) of the net if, for every neighbourhood of and every , 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.

## Selected facts

- We have the following characterization 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 neighborhood of the point meets the set. Now,*x*is a limit point of*S*, if and only if every neighborhood of*x*contains a point of*S*other than*x*, if and only if every neighborhood 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 characterization of the closure of*S*: The closure of*S*is equal to the union of*S*and L(*S*). This fact is sometimes taken as the*definition*of closure.*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*. If*x*is not in*S*, then the complement to*S*comprises an open neighbourhood of*x*. Since*x*is a limit point of*S*, any open neighbourhood of*x*should have a non-trivial intersection with*S*. However, a set can not have a non-trivial intersection with its complement. 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 neighbourhood*U*of*x*that does not intersect*S*, and so*U*lies entirely in the complement of*S*. Since this argument holds for arbitrary*x*in the complement of*S*, the complement of*S*can be expressed as a union of open neighbourhoods of the points 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*.

- The closure
*cl(S)*of a set*S*is a disjoint union of its limit points*L(S)*and isolated points*I(S)*:

- 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*.

- By definition, every limit point is an adherent point.

## References

- Hazewinkel, Michiel, ed. (2001) [1994], "Limit point of a set",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4