View source for Topology induced by a metric

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==Definition==
Let {{M|(X,d)}} be a [[metric space]]. Then there is a [[topology]] we can imbue on {{M|X}}, called the ''metric topology'' that can be defined in terms of the [[metric]], {{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }}.

We do this using the concept of [[topology generated by a basis]]. We claim ("'''''Claim 1'''''"):
* {{M|\mathcal{B}:\eq\left\{B_r(x)\ \vert\ x\in X\wedge r\in\mathbb{R}_{>0} \right\} }} satisfies the conditions to generate a [[topology]] (and is a basis for that topology) - where {{M|B_\epsilon(p)}}<ref group="Note">For convenience I restate this now:
* {{M|B_\epsilon(p):\eq\{ x\in X\ \vert d(x,p)<\epsilon\} }}
Notice that {{M|p\in B_\epsilon(p)}} always as {{M|d(p,p)\eq 0<\epsilon}} so {{M|p\in B_\epsilon(p)}}</ref> denotes the [[open ball]] of radius {{M|\epsilon\in\mathbb{R}_{>0} }} centred at {{M|p\in X}}
The resulting [[topological space]], say {{Top.|X|J}}, has [[topological basis|basis]] {{M|\mathcal{B} }}
* Explicitly the [[topology]] is {{MM|\mathcal{J}:\eq\left\{\left. \bigcup_{B\in\mathcal{F} }B\ \right\vert\ \mathcal{F}\in\mathcal{P}(\mathcal{B})\right\} }}
** Notice {{M|\bigcup_{B\in\emptyset} B\eq\emptyset}} - hence the [[empty-set]] is [[open set|open]] - as required.
** Notice also that {{M|\bigcup{B\in\mathcal{B} }B\eq X}} - obvious as {{M|\mathcal{B} }} contains (among others) an open ball centred at each point in {{M|X}} and each point is in that open ball at least. {{XXX|Copy and paste a proof from elsewhere}}
==Consequences==
* The [[open sets]] are exactly those which are {{plural|union|s}} of basis elements. {{ie}} 

{{Requires work|grade=A*|msg=These should have more far-reaching consequences on the site. Like on the [[neighbourhood]] page and such}}
==Proof of claims==
* '''Claim 1: {{M|\mathcal{B} }} satisfies the conditions to generate a topology for which it is a basis''' - see ''[[The set of all open balls of a metric space are able to generate a topology and are a basis for that topology]]''
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Metric Space|Topology}}
{{Theorem Of|Metric Space|Topology}}