Metric space

A normed space is a special case of a metric space, to see the relationships between metric spaces and others see: Subtypes of topological spaces

Definition of a metric space

A metric space is a set

$X$ coupled with a "distance function"^[1]:

With the properties that for

$x,y,z\in X$ :

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

$X$

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:

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

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Proof that this is a metric

This is a useless metric, but is a metric and induces the Discreet Topology on X, where the topology is the powerset of

$X$ ,

$\mathcal{P}(X)$ .

It is given by:

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

Note: it is sometimes called the trivial metric^[3]

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Proof that this is a metric