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
Contents
Definition of a metric space
A metric space is a set [math]X[/math] coupled with a "distance function"^{[1]}^{[2]}:
 [math]d:X\times X\rightarrow\mathbb{R}[/math] or sometimes
 [math]d:X\times X\rightarrow\mathbb{R}_+[/math]^{[3]}, Note that here I prefer the notation [math]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/math]
With the properties that for [math]x,y,z\in X[/math]:
 [math]d(x,y)\ge 0[/math] (This is implicit with the [ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/ilmath] definition)
 [math]d(x,y)=0\iff x=y[/math]
 [math]d(x,y)=d(y,x)[/math]  Symmetry
 [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}_{\ge 0})[/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{\sum^n_{i=1}((x_iy_i)^2)}[/math]
Proof that this is a metric
TODO:
Discrete Metric
Let [ilmath]X[/ilmath] be a set. The discrete^{[2]} metric, or trivial metric^{[4]} is the metric defined as follows:
 [math]d:X\times X\rightarrow \mathbb{R}_{\ge 0} [/math] with [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right.[/math]
However any strictly positive value will do for the [ilmath]x\ne y[/ilmath] case. For example we could define [ilmath]d[/ilmath] as:
 [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.[/math]
 Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath]^{[Note 1]}  traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.
 Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath]^{[Note 1]}  traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.
Note: however in proofs we shall always use the case [ilmath]v=1[/ilmath] for simplicity
Notes
Property  Comment 

induced topology  discrete topology  which is the topology [ilmath](X,\mathcal{P}(X))[/ilmath] (where [ilmath]\mathcal{P} [/ilmath] denotes power set) 
Open ball  [ilmath]B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right.[/ilmath] 
Open sets  Every subset of [ilmath]X[/ilmath] is open. Proof outline: as for a subset [ilmath]A\subseteq X[/ilmath] we can show [ilmath]\forall x\in A\exists r[B_r(x)\subseteq A][/ilmath] by choosing say, that is [ilmath]A[/ilmath] contains an open ball centred at each point in [ilmath]A[/ilmath]. 
Connected  The topology generated by [ilmath](X,d_\text{discrete})[/ilmath] is not connected if [ilmath]X[/ilmath] has more than one point. Proof outline:

See also
Notes
 ↑ Note the strictly greater than 0 requirement for [ilmath]v[/ilmath]
References
 ↑ Introduction to Topology  Bert Mendelson
 ↑ ^{2.0} ^{2.1} Introduction to Topology  Theodore W. Gamelin & Robert Everist Greene
 ↑ Analysis  Part 1: Elements  Krzysztof Maurin
 ↑ Functional Analysis  George Bachman and Lawrence Narici