Real coordinate space

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$ n\in \mathbb N $

$\mathbb R^n$

Recursive definition:

$ p\in \mathbb N $

$ p>1 $

$ \mathbb R^1=\mathbb R $

$ \mathbb R^p=\mathbb R^{p-1}\times \mathbb R, \hspace{1cm}$

$1\le i\le p$

$ a< b \equiv \forall i.\ a_i< b_i$

$ a> b \equiv \forall i.\ a_i> b_i$

$ a\le b \equiv \forall i.\ a_i\le b_i$

$ a\ge b \equiv \forall i.\ a_i\ge b_i$

$ [a,b]\equiv\{x\ |\ \forall i.\ a_i\le x_i\le b_i\}$

$ ]a,b]\equiv\{x\ |\ \forall i.\ a_i<x_i\le b_i\}$

$ [a,b[\ \equiv\{x\ |\ \forall i.\ a_i\le x_i<b_i\}$

$ ]a,b[\ \equiv\{x\ |\ \forall i.\ a_i<x_i<b_i\}$