Metric Space

analysis topology

Definition

Metric Space

A set $M$ equipped with a mapping $d:M\times M\to \mathbb{R}$, called a metric, is a metric space $⟨M,d⟩$ if the following axioms hold for all $x,y,z\in M$:

1. Non-negativity: $d(x,y)\ge 0$
2. Identity of Indiscernibles: $d(x,y)=0⟺x=y$
3. Symmetry: $d(x,y)=d(y,x)$
4. Triangle Inequality: $d(x,y)\le d(x,z)+d(z,y)$

Visual Intuition

The triangle inequality states that the direct distance between two points is never greater than the sum of the distances through an intermediate point.

Common Metrics in ${\mathbb{R}}^n$

The following are standard metrics for vectors $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$:

• $L_p$ Metric:$d_p(\mathbf{x},\mathbf{y})={\left({\sum }_{i=1}^n|x_i-y_i{|}^p\right)}^{1/p}$ for $1\le p<\infty$.
• Sum Metric ($L_1$):$d_1(\mathbf{x},\mathbf{y})={\sum }_{i=1}^n|x_i-y_i|$.
• Euclidean Metric ($L_2$):$d_2(\mathbf{x},\mathbf{y})=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$.
• Max Metric ($L_{\infty }$):$d_{\infty }(\mathbf{x},\mathbf{y})={\max }_{1\le i\le n}|x_i-y_i|$.
• Hamming Distance: $d_H(\mathbf{x},\mathbf{y})=|\{i\mid x_i\ne y_i\}|$, primarily used for discrete spaces.

Neighbourhoods

A neighbourhood (or open ball) in a metric space is the set of all points within a certain distance from a centre point:

Open Ball

Given $x\in M$ and a radius $r>0$, the open ball $B_r(x)$ is defined as:

$$B_r(x)=\{y\in M\mid d(x,y)<r\}$$

This concept is used to generalise epsilon neighbourhoods to arbitrary metric spaces.