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 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 definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is not synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter ^{ [1] } (respectively, a limit point of a sequence,^{ [2] } a limit point of a net) refers to a point that the filter converges to (respectively, the sequence converges to, the net converges to).
The limit points of a set should not be confused with adherent points for which every neighbourhood of contains a point of . Unlike for limit points, this point of may be itself. A limit point can be characterized as an adherent point that is not an isolated point.
Limit points of a set should also not be confused with boundary points. For example, is a boundary point (but not a limit point) of set in with standard topology. However, is a limit point (though not a boundary point) of interval in with standard topology (for a less trivial example of a limit point, see the first caption).^{ [3] }^{ [4] }^{ [5] }
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.
Let be a subset of a topological space A point in is a limit point or cluster point or accumulation point of a set of if every neighbourhood of contains at least one point of different from itself.
It does not 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 space (such as a metric space), then is a limit point of if and only if every neighbourhood of contains infinitely (distinct) many points of ^{ [6] } In fact, spaces are characterized by this property.
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
If every neighbourhood of contains infinitely many points of then is a specific type of limit point called an ω-accumulation point of
If every neighbourhood of contains uncountably many points of then is a specific type of limit point called a condensation point of
If every neighbourhood of satisfies then is a specific type of limit point called a complete accumulation point of
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 a 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 filters.
Every sequence in is by definition just a map so that its image can be defined in the usual way.
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 satisfy
Every limit of a non-constant sequence is an accumulation point of the sequence. And by definition, every limit point is an adherent point.
The closure of a set is a disjoint union of its limit points and isolated points :
A point is a limit point of if and only if it is in the closure of
Proof |
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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 |
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("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 |
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Proof 1: is closed if and only if is equal to its closure if and only if if and only if is contained in Proof 2: 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 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 |
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If is an isolated point, then is a neighbourhood of that contains no points other than |
A space is discrete if and only if no subset of has a limit point.
Proof |
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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 |
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As long as is nonempty, its closure will be It is only empty when is empty or is the unique element of |
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