Topology
- Preliminaries and Introduction
- Convergence in Topology
- Introduction to Normed Spaces
- Vector Spaces & Linear Operators
- Finite Dimensional Normed Spaces
- Infinite Dimensional Normed Spaces
- Compactness and Completeness in Normed Spaces
Convergence in Topology
Sequence Convergence in Metric Spaces
Set equipped with a metric (distance function).
In the study of topology, a key concept is the convergence of sequences in metric spaces. This article will provide a comprehensive overview of this topic, covering the definition of a sequence in a metric space, the convergence of sequences, properties of convergent sequences, subsequences, and limit points.
Definition of a Sequence in a Metric Space
A metric space is a set equipped with a function (the metric) that measures the distance between every pair of elements in the set. A sequence in a metric space is an ordered infinite list of elements from the space. Formally, a sequence in a metric space (M, d) is a function from the set of natural numbers N to M.
Convergence of Sequences in Metric Spaces
A sequence in a metric space is said to converge if, as you go further and further along the sequence, the elements get arbitrarily close to a certain value, known as the limit of the sequence. Formally, a sequence (x_n) in a metric space (M, d) converges to a limit x in M if for every positive real number ε, there exists a natural number N such that for all n > N, the distance d(x_n, x) < ε.
Properties of Convergent Sequences
Convergent sequences in metric spaces have several important properties. For example, every convergent sequence is bounded, meaning there is some number M such that the distance from every term in the sequence to the limit is less than M. Additionally, the limit of a convergent sequence is unique.
Subsequences and Their Properties
A subsequence of a sequence is a sequence formed by taking elements from the original sequence in their original order, but not necessarily all of them. If a sequence converges, then every subsequence also converges, and they all converge to the same limit.
Limit Points and Isolated Points
A limit point of a set in a metric space is a point such that every open ball centered at that point contains infinitely many points from the set. If a sequence in a metric space has a limit, then that limit is a limit point of the set of values of the sequence. An isolated point of a set is a point that is not a limit point.
In conclusion, understanding sequence convergence in metric spaces is fundamental to the study of topology. It provides the groundwork for more advanced topics such as Cauchy sequences and point-set topology.