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
Understanding Cauchy Sequences
Sequence whose elements become arbitrarily close to each other.
In the realm of mathematical analysis, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. This concept is fundamental to understanding convergence in metric spaces, and it plays a crucial role in the study of normed spaces.
Definition and Properties of Cauchy Sequences
A sequence (x_n) in a metric space (X, d) is called a Cauchy sequence if for every positive real number ε, there exists a positive integer N such that for all natural numbers m, n > N, the distance d(x_m, x_n) < ε. In simpler terms, the elements of a Cauchy sequence become arbitrarily close to each other as the sequence progresses.
Relationship Between Cauchy Sequences and Convergent Sequences
In a complete metric space, a sequence is convergent if and only if it is a Cauchy sequence. However, this is not true for all metric spaces. There exist metric spaces where Cauchy sequences do not converge. These are known as incomplete metric spaces.
Completeness of a Metric Space
A metric space is said to be complete if every Cauchy sequence in that space converges to a limit in the same space. The real numbers with the usual metric (absolute difference) is an example of a complete metric space. On the other hand, the rational numbers with the usual metric is an example of an incomplete metric space, as there exist Cauchy sequences of rational numbers that do not converge to a rational number (for example, the sequence of decimal approximations to the square root of 2).
Examples of Complete and Incomplete Metric Spaces
As mentioned above, the set of real numbers (R, |.|) and the set of complex numbers (C, |.|) are examples of complete metric spaces. The set of rational numbers (Q, |.|) is an example of an incomplete metric space.
The Cauchy Completion of a Metric Space
Given an incomplete metric space, it is possible to "complete" it by adding in the limits of all Cauchy sequences. This process is known as the Cauchy completion of a metric space. The completion of the rational numbers (Q, |.|) is the real numbers (R, |.|).
Understanding Cauchy sequences is crucial for the study of normed spaces, as it allows us to discuss convergence and completeness in these spaces. In the next unit, we will explore point-set topology, which will further expand our understanding of convergence in topological spaces.