The real numbers

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• Be sure to include Example:The real line with the finite complement topology is not Hausdorff

The real numbers
$\mathbb{R}$
Algebraic structure
TODO: Todo - is a field
Standard topological structures
Main page: The real line
inner product	$\langle a,b\rangle:\eq a*b$ - Euclidean inner-product on $\mathbb{R}^1$
norm	$\Vert x\Vert:\eq\sqrt{\langle x,x\rangle}\eq\vert x\vert$ - Euclidean norm on $\mathbb{R}^1$
metric	$d(x,y):\eq\Vert x-y\Vert\eq \vert x-y\vert$ - Absolute value - Euclidean metric on $\mathbb{R}^1$
topology	topology induced by the metric $d$
Standard measure-theoretic structures
measurable space	Borel $\sigma$-algebra of $\mathbb{R}$
- other:	Lebesgue-measurable sets of $\mathbb{R}$ contains the Borel $\sigma$-algebra

• 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 sigma-algebra of the real line - useful in Measure Theory although distinct from Lebesgue measurable sets on the real line
  • TODO: Pages needed
    for the Lebesgue-measurable structure on $\mathbb{R}^n$ and $\mathbb{R}$

Definition

Cantor's construction of the real numbers

The set of real numbers, $\mathbb{R}$, is the quotient space, $\mathscr{C}/\sim$ where:

• $\mathscr{C}$ - the set of all Cauchy sequences in $\mathbb{Q}$ - the quotients
• $\sim$ - the usual equivalence of Cauchy sequences

We further claim:

1. that the familiar operations of addition, multiplication and division are well defined and
2. by associating $x\in\mathcal{Q}$ with the sequence $({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q}$ where $\forall n\in\mathbb{N}[x_n:=x]$ we can embed $\mathbb{Q}$ in $\mathbb{R}:=\mathscr{C}/\sim$

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

$\mathbb{R}$ is an example of:

• Vector space
• Field ($\implies\ \ldots\implies$ ring)
• Complete metric space ($\implies$ topological space)
  • With the metric of absolute value

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TODO: Flesh out

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Properties

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• The axiom of completeness - a badly named property that isn't really an axiom.

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If $S\subseteq\mathbb{R}$ is a non-empty set of real numbers that has an upper bound then:

• $\text{Sup}(S)$ (the supremum of $S$) exists.