Metric space

Definition of a metric space

A metric space is a set $$X$$ coupled with a "distance function" $$d:X\times X\rightarrow\mathbb{R}$$ with the properties (for $$x,y,z\in X$$ )

1. $$d(x,y)\ge 0$$
2. $$d(x,y)=0\iff x=y$$
3. $$d(x,y)=d(y,x)$$
4. $$d(x,z)\le d(x,y)+d(y,z)$$ - the Triangle Inequality

We will denote a metric space as $$(X,d)$$ (as $$(X,d:X\times X\rightarrow\mathbb{R})$$ is too long and Mathematicians are lazy) or simply $$X$$ if it is obvious which metric we are talking about on $$X$$

Examples of metrics

Euclidian Metric

The Euclidian metric on $$\mathbb{R}^n$$ is defined as follows: For $$x=(x_1,...,x_n)\in\mathbb{R}^n$$ and $$y=(y_1,...,y_n)\in\mathbb{R}^n$$ we define the Euclidian metric by:

$$d_{\text{Euclidian}}(x,y)=\sqrt{\prod^n_{i=1}(x_i^2+y_i^2)}$$

Proof it is a metric

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TODO: Proof this is a metric

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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 $$X$$ , $$\mathcal{P}(X)$$ .

$$d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr} 1 & x=y\\ 0 & \text{otherwise} \end{array}\right.$$