Everything about Limit Point totally explained
In
mathematics, informally speaking, a
limit point of a set
S in a
topological space X is a point
x in
X that can be "approximated" by points of
S other than
x as well as one pleases. 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 adding its limit points.
A related concept is
cluster point or
accumulation point of a
sequence.
Definition
Let
S be a subset of a
topological space X.
We say that a point
x in
X is a
limit point of
S if
every
open set containing
x also contains a point of
S other than
x itself. This is equivalent to requiring that every
neighbourhood of
x contains a point of
S other than
x itself. (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.)
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 a
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.
Cluster point
If
X is a
metric space with distance
d, then a point
x in
X is a
cluster point or
accumulation point of a
sequence (
xn ) if for every
ε > 0, there are infinitely many values of
n such that
d (
x,
xn ) <
ε. Equivalently, that every
neighborhood of
x contains
xn for infinitely many
n.
A limit point of the set of points in a sequence is a cluster point of the sequence. However, if for infinitely many
n the values of
xn are equal, this point is a cluster point of the sequence but not necessarily a limit point of the set of points in the sequence.
A cluster point of a sequence is a
subsequential limit: the limit of some
subsequence.
The concept of a
net generalizes the idea of a sequence. Cluster points in nets encompass the idea of both condensation points and ω-accumulation points:
If φ is a net on
X based on directed set
D and
A is a subset of
X, then φ is frequently in
A if for every α in
D there exists some β ≥ α, β in
D, so that φ(β) is in
A. A point
x in
X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood
U of
x, the net is frequently in
U.
Clustering and limit points are also defined for the related topic of
filters.
The set of all cluster points of a sequence is sometimes called a
limit set.
Some facts
- We have the following characterisation of limit points: x is a limit point of S if and only if it's in the closure of S converging to L.
Further Information
Get more info on 'Limit Point'.
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