9. Complete Spaces and Uniform Spaces

Complete Spaces and Uniform Spaces

Completeness in Metric Spaces

Completeness: A metric space \( (X, d) \) is said to be complete if every Cauchy sequence in \( X \) converges to a limit in \( X \). That is, for every sequence \( (x_n) \) in \( X \), if for every \( \varepsilon > 0 \) there exists an \( N \) such that for all \( m, n > N \), \( d(x_m, x_n) < \varepsilon \), then there exists \( x \) in \( X \) such that \( \lim_{n \to \infty} x_n = x \).

Examples:

• The real numbers \( \mathbb{R} \) with the standard Euclidean metric is a complete metric space.
• The rational numbers \( \mathbb{Q} \) with the standard Euclidean metric is not complete.

Definition of Uniform Spaces

Uniform Space: A uniform space is a set equipped with a uniform structure, which generalizes the notion of distance in metric spaces. A uniform structure consists of a collection of entourages satisfying certain axioms.

In a uniform space \( X \), an entourage is a subset of \( X \times X \) that "uniformly covers" pairs of points in \( X \). For example, if \( U \) is an entourage, then for every pair of points \( (x, y) \) in \( X \), there exists a subset \( V \) of \( X \) such that \( (x, y) \) is contained in \( V \times V \) and \( V \times V \) is a subset of \( U \).

Examples of Complete and Incomplete Spaces

Complete Space: The real numbers \( \mathbb{R} \) with the standard Euclidean metric is a complete metric space.

Incomplete Space: The space of continuous functions on a closed interval \( [a, b] \) with the supremum metric (also known as the uniform metric) is incomplete. This can be shown by considering the sequence of polynomial functions that converges pointwise to a non-polynomial function.