Accumulation point

There is also a closely related concept for sequences.

This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.

The similarly named notion of a limit point of a sequence[1] (respectively, a limit point of a filter,[2] a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to).

Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters).

That is, the term "limit point of a sequence" is not synonymous with "cluster/accumulation point of a sequence".

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.

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.

contains infinitely many points of

In fact, Fréchet–Urysohn spaces are characterized by this property.

contains infinitely many points of

is a specific type of limit point called a complete accumulation point of

Note that there is already the notion of limit of a sequence to mean a point

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.

Clustering and limit points are also defined for filters.

Conversely, given a countable infinite set

in many ways, even with repeats, and thus associate with it many sequences

is a disjoint union of its limit points

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.

to denote the set of limit points of

This fact is sometimes taken as the definition of closure.

is closed if and only if it contains all of its limit points.

However, a set can not have a non-trivial intersection with its complement.

is not a limit point, and hence there exists an open neighbourhood

Since this argument holds for arbitrary

can be expressed as a union of open neighbourhoods of the points in the complement of