4. Topology of Metric Spaces

Topology of Metric Spaces

Metric Spaces and Their Topology

Metric Space: A metric space is a set \( X \) equipped with a distance function \( d: X \times X \rightarrow \mathbb{R} \) satisfying the following properties for all \( x, y, z \) in \( X \):

1. Non-negativity: \( d(x, y) \geq 0 \) with equality if and only if \( x = y \).
2. Identity of indiscernibles: \( d(x, y) = 0 \) if and only if \( x = y \).
3. Symmetry: \( d(x, y) = d(y, x) \).
4. Triangle inequality: \( d(x, z) \leq d(x, y) + d(y, z) \).

The topology induced by the metric \( d \) consists of open sets defined by open balls \( B(x, r) = \{ y \in X \mid d(x, y) < r \} \) for every \( x \) in \( X \) and \( r > 0 \).

Properties of Metric Spaces

Completeness: A metric space \( X \) is complete if every Cauchy sequence in \( X \) converges to a limit in \( X \).

Compactness: A metric space \( X \) is compact if every open cover of \( X \) has a finite subcover.

Connectedness: A metric space \( X \) is connected if it cannot be divided into two disjoint nonempty open sets.

Examples of Metric Spaces

Euclidean Space: The Euclidean space \( \mathbb{R}^n \) equipped with the standard Euclidean metric is a metric space. The distance between two points \( x = (x_1, x_2, \ldots, x_n) \) and \( y = (y_1, y_2, \ldots, y_n) \) is given by \( d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2} \).

Discrete Metric Space: Any set \( X \) equipped with the discrete metric \( d(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases} \) is a metric space.