Math and science::Topology

Metric space

Metric

A metric $d$ on a set $X$ is a function $d : X \times X \to [0, \infty)$ with the following three properties:

What is this called?: $d (x, y) = 0 ⟺ x = y, for all x, y \in X$
Triangle inequality: $d (x, y) + d (y, z) \geq d (x, z), for all x, y, z \in X$
Symmetry: $d (x, y) = d (y, x), for all x, y \in X$

A metric space is a set together with a metric on it, or more formally, a pair $(X, d)$ where $X$ is a set and $d$ is a metric on $X$ .

Example

Some interesting metrics

1. $d_{1}, d_{2}, d_{\infty}$ (instances of $d_{p}$ ) on $R^{n}$

For all $x = (x_{1}, . . ., x_{n})$ and $y = (y_{1}, . . ., y_{n})$ in $R^{n}$ (or any subset thereof):

The Euclidean metric, $d_{2}$ on $R^{n}$ is given by: $d_{2} (x, y) = (\sum_{i = 1}^{n} (x_{i} - y_{i})^{2})^{\frac{1}{2}}$
$d_{1}$ is given by: $d_{1} (x, y) = \sum_{i = 1}^{n} | x_{i} - y_{i} |$
$d_{\infty}$ by: $d_{\infty} (x, y) = m a x_{1 \leq i \leq n} | x_{i} - y_{i} |$

All these metrics are particular cases of the $d_{p}$ metric.

2. $d_{1}, d_{2}, d_{\infty}$ (instances of $d_{p}$ ) on $C [a, b]$ )

All $d_{p}$ metrics have parallels over function spaces. $C [a, b]$ is the set of continuous functions $[a, b] \to R$ . Three interesting metrics on $C [a, b]$ are given by:

$d_{1} (f, g) = \int_{a}^{b} | f (t) - g (t) | d t$

$d_{2} (f, g) = {(\int_{a}^{b} | f (t) - g (t) | d t)}^{\frac{1}{2}}$

$d_{\infty} (f, g) = sup_{a \leq t \leq b} | f (t) - g (t) |$

3. Hamming metric

With $A$ being any set, and $x, y \in A^{n}$ , the Hamming metric $d$ on $A$ is given by

$d (x, y) = | {i \in {1, . . ., n} : x_{i} \neq y_{i}} |$

In other words, the count of how many coordinates differ.

Metric space

Metric

Example

1. d1,d2,d∞ (instances of dp) on Rn

2. d1,d2,d∞ (instances of dp) on C[a,b])

3. Hamming metric

Context

1. $d_{1}, d_{2}, d_{\infty}$ (instances of $d_{p}$ ) on $R^{n}$

2. $d_{1}, d_{2}, d_{\infty}$ (instances of $d_{p}$ ) on $C [a, b]$ )