Metric space
Contents
Definition of a metric space
A metric space is a set [math]X[/math] coupled with a "distance function" [math]d:X\times X\rightarrow\mathbb{R}[/math] with the properties (for [math]x,y,z\in X[/math])
- [math]d(x,y)\ge 0[/math]
- [math]d(x,y)=0\iff x=y[/math]
- [math]d(x,y)=d(y,x)[/math]
- [math]d(x,z)\le d(x,y)+d(y,z)[/math] - the Triangle Inequality
We will denote a metric space as [math](X,d)[/math] (as [math](X,d:X\times X\rightarrow\mathbb{R})[/math] is too long and Mathematicians are lazy) or simply [math]X[/math] if it is obvious which metric we are talking about on [math]X[/math]
Examples of metrics
Euclidian Metric
The Euclidian metric on [math]\mathbb{R}^n[/math] is defined as follows: For [math]x=(x_1,...,x_n)\in\mathbb{R}^n[/math] and [math]y=(y_1,...,y_n)\in\mathbb{R}^n[/math] we define the Euclidian metric by:
[math]d_{\text{Euclidian}}(x,y)=\sqrt{\prod^n_{i=1}(x_i^2+y_i^2)}[/math]
Proof it is a metric
TODO: Proof this is a metric
Discreet Metric
This is a useless metric, but is a metric and induces the Discreet Topology on X, where the topology is the powerset of [math]X[/math], [math]\mathcal{P}(X)[/math].
[math]d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr} 1 & x=y\\ 0 & \text{otherwise} \end{array}\right.[/math]
