Basics of Metric Spaces

Set equipped with a metric (distance function).

A metric space is a set where a notion of distance (called a metric) is defined. The metric allows us to measure the distance between any two points in the set. Metric spaces are a key concept in the field of topology and are used to study continuity, convergence, and more.

Definition and Examples of Metric Spaces

A metric space is an ordered pair (X, d) where X is a set and d is a metric on X. The metric, or distance function, is a function d: X × X → R (where R is the set of real numbers) that satisfies the following conditions for all x, y, and z in X:

1. d(x, y) ≥ 0 (non-negativity)
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles)
3. d(x, y) = d(y, x) (symmetry)
4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Examples of metric spaces include the set of real numbers with the standard metric (absolute difference), the Euclidean space (with the Euclidean distance), and the set of continuous functions on a closed interval [a, b] (with the supremum metric).

Open and Closed Balls in Metric Spaces

In a metric space (X, d), for any point x in X and any real number r > 0, we can define the open ball B(x, r) and the closed ball B[x, r] as follows:

• B(x, r) = {y in X : d(x, y) < r}
• B[x, r] = {y in X : d(x, y) ≤ r}

These balls are fundamental in defining open and closed sets in metric spaces.

Properties of Metric Spaces: Completeness, Compactness, and Connectedness

A metric space is said to be complete if every Cauchy sequence in X converges to a limit that is also in X. Compactness in metric spaces is equivalent to sequential compactness, meaning every sequence has a subsequence that converges to a limit in the space. A metric space is connected if it cannot be divided into two nonempty open sets.

Continuity in Metric Spaces

A function f: X → Y between two metric spaces (X, dX) and (Y, dY) is continuous at a point x in X if for every positive real number ε, there exists a positive real number δ such that for all points x' in X, if dX(x, x') < δ then dY(f(x), f(x')) < ε. This is the ε-δ definition of continuity in the context of metric spaces.

Convergence in Metric Spaces

A sequence (x_n) in a metric space (X, d) is said to converge to a limit x in X if for every positive real number ε, there exists a positive integer N such that for all integers n > N, we have d(x_n, x) < ε. This is the ε-N definition of convergence in the context of metric spaces.

Understanding the basics of metric spaces is crucial for studying more advanced topics in topology, such as normed spaces, which will be covered in later units.