Real coordinate space

Set

context $ n\in \mathbb N $
definiendum $\mathbb R^n$

Discussion

Explicitly,

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

range $1\le i\le p$
predicate $ a< b \equiv \forall i.\ a_i< b_i$
predicate $ a> b \equiv \forall i.\ a_i> b_i$
predicate $ a\le b \equiv \forall i.\ a_i\le b_i$
predicate $ a\ge b \equiv \forall i.\ a_i\ge b_i$
todo: define the following in a seperate entry:

$ [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\}$

Reference

Wikipedia: Real coordinate space

Parents

Subset of

Cartesian product

Requirements

Real number