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 point
If every neighborhood of contains infinitely many points of , then is a specific type of limit point called an ω-accumulation point of .
If every neighborhood of satisfies , then is a specific type of limit point called a complete accumulation point of .
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.
Note that there is already the notion of limit of a sequence to mean a point to which the sequence converges (that is, every neighborhood of contains all but finitely many elements of the sequence). That is why we do not use the term limit point of a sequence as a synonym for accumulation point of the sequence.
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.
- Every accumulation point of a sequence is an accumulation point of any subsequence of the sequence.
- Every limit of a sequence is an accumulation point of the sequence.
Relation between accumulation point of a sequence and accumulation point of a set
To each sequence in a topological space we can associate the set consisting of all the elements in the sequence.
- If there a element that occurs infinitely many times in the sequence, is an accumulation point of the sequence. But need not be an accumulation point of the corresponding set . For example, if the sequence is the constant sequence with value , we have and is an isolated point of and not an accumulation point of .
- If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an -accumulation point of the associated set .
Conversely, given a countable infinite set in , we can enumerate all the elements of in many ways, even with repeats, and thus associate with it many sequences that will have as associated set of elements.
- Any -accumulation point of is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of and hence also infinitely many terms in any associated sequence).
- A point that is not an -accumulation point of cannot be an accumulation point of any of the associated sequences without infinite repeats (because has a neighborhood that contains only finitely many (even none) points of and that neighborhood can only contain finitely many terms of such sequences).
- 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 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.