In mathematics, a limit point (or cluster point or accumulation point) of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of .
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 in a topological space is a point such that, for every neighbourhood of , there are infinitely many natural numbers such that . This concept generalizes to nets and filters.
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 is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then is a limit point of if and only if there is a sequence of points in whose limit is . In fact, Fréchet–Urysohn spaces are characterized by this property.
The set of limit points of is called the derived set of .
Types of limit pointsEdit
If every open set containing contains infinitely many points of , then is a specific type of limit point called an -accumulation point of .
If every open set containing satisfies , then is a specific type of limit point called a complete accumulation point of .
For sequences and netsEdit
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.
- We have the following characterization of limit points: is a limit point of if and only if it is in the closure of .
- 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, is a limit point of , if and only if every neighborhood of contains a point of other than , if and only if every neighborhood of contains a point of , if and only if is in the closure of .
- If we use to denote the set of limit points of , then we have the following characterization of the closure of : The closure of is equal to the union of and . This fact is sometimes taken as the definition of closure.
- Proof: ("Left subset") Suppose is in the closure of . If is in , we are done. If is not in , then every neighbourhood of contains a point of , and this point cannot be . In other words, is a limit point of and is in . ("Right subset") If is in , then every neighbourhood of clearly meets , so is in the closure of . If is in , then every neighbourhood of contains a point of (other than ), so is again in the closure of . This completes the proof.
- A corollary of this result gives us a characterisation of closed sets: A set is closed if and only if it contains all of its limit points.
- Proof: is closed if and only if is equal to its closure if and only if if and only if is contained in .
- Another proof: Let be a closed set and a limit point of . If is not in , then the complement to comprises an open neighbourhood of . Since is a limit point of , any open neighbourhood of should have a non-trivial intersection with . However, a set can not have a non-trivial intersection with its complement. Conversely, assume contains all its limit points. We shall show that the complement of is an open set. Let be a point in the complement of . By assumption, is not a limit point, and hence there exists an open neighbourhood U of that does not intersect , and so lies entirely in the complement of . Since this argument holds for arbitrary in the complement of , the complement of can be expressed as a union of open neighbourhoods of the points in the complement of . Hence the complement of is open.
- No isolated point is a limit point of any set.
- Proof: If is an isolated point, then is a neighbourhood of that contains no points other than .
- The closure of a set is a disjoint union of its limit points and isolated points :
- A space is discrete if and only if no subset of has a limit point.
- Proof: If is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if is not discrete, then there is a singleton that is not open. Hence, every open neighbourhood of contains a point , and so is a limit point of .
- If a space has the trivial topology and is a subset of with more than one element, then all elements of are limit points of . If is a singleton, then every point of is still a limit point of .
- Proof: As long as } is nonempty, its closure will be . It's only empty when is empty or is the unique element of .
- By definition, every limit point is an adherent point.