The real numbers
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The real numbers  
[ilmath]\mathbb{R} [/ilmath]
 
Algebraic structure  

TODO: Todo  is a field
 
Standard topological structures  
Main page: The real line  
inner product  [ilmath]\langle a,b\rangle:\eq a*b[/ilmath]  Euclidean innerproduct on [ilmath]\mathbb{R}^1[/ilmath] 
norm  [ilmath]\Vert x\Vert:\eq\sqrt{\langle x,x\rangle}\eq\vert x\vert[/ilmath]  Euclidean norm on [ilmath]\mathbb{R}^1[/ilmath] 
metric  [ilmath]d(x,y):\eq\Vert xy\Vert\eq \vert xy\vert[/ilmath]  Absolute value  Euclidean metric on [ilmath]\mathbb{R}^1[/ilmath] 
topology  topology induced by the metric [ilmath]d[/ilmath] 
Standard measuretheoretic structures  
measurable space  Borel [ilmath]\sigma[/ilmath]algebra of [ilmath]\mathbb{R} [/ilmath]^{[Note 1]} 

Lebesguemeasurable sets of [ilmath]\mathbb{R} [/ilmath]

 The real line is the name given to the reals with their "usual topology", the topology that is induced by the absolute value metric
 Borel sigmaalgebra of the real line  useful in Measure Theory although distinct from Lebesgue measurable sets on the real line
 TODO: Pages neededfor the Lebesguemeasurable structure on [ilmath]\mathbb{R}^n[/ilmath] and [ilmath]\mathbb{R} [/ilmath]

Contents
Definition
Cantor's construction of the real numbers
The set of real numbers, [ilmath]\mathbb{R} [/ilmath], is the quotient space, [ilmath]\mathscr{C}/\sim[/ilmath] where:^{[1]}
 [ilmath]\mathscr{C} [/ilmath]  the set of all Cauchy sequences in [ilmath]\mathbb{Q} [/ilmath]  the quotients
 [ilmath]\sim[/ilmath]  the usual equivalence of Cauchy sequences
We further claim:
 that the familiar operations of addition, multiplication and division are well defined and
 by associating [ilmath]x\in\mathcal{Q} [/ilmath] with the sequence [ilmath] ({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q} [/ilmath] where [ilmath]\forall n\in\mathbb{N}[x_n:=x][/ilmath] we can embed [ilmath]\mathbb{Q} [/ilmath] in [ilmath]\mathbb{R}:=\mathscr{C}/\sim[/ilmath]
Axiomatic construction of the real numbers
Axiomatic construction of the real numbers/Definition
[ilmath]\mathbb{R} [/ilmath] is an example of:
 Vector space
 Field ([ilmath]\implies\ \ldots\implies[/ilmath] ring)
 Complete metric space ([ilmath]\implies[/ilmath] topological space)
 With the metric of absolute value
TODO: Flesh out
Properties
 The axiom of completeness  a badly named property that isn't really an axiom.
If [ilmath]S\subseteq\mathbb{R} [/ilmath] is a nonempty set of real numbers that has an upper bound then^{[2]}:
 [ilmath]\text{Sup}(S)[/ilmath] (the supremum of [ilmath]S[/ilmath]) exists.
Notes
 ↑ This is just the Borel sigmaalgebra on the real line (with its usual topology)