[[Mathematics MOC]]
# Real coördinate space

**Real coördinate space** $\mathbb{R}^n$ is the set of $n$-tuples of [[Real numbers]].

## Topological properties

Considered as either a [[metric topology]] or [[product topology]], $\mathbb{R}^n$ has the following properties

- [[Heine-Borel theorem]]: Subsets compact iff closed and bounded.
- $\mathbb{R}^n$ is [[Path connectedness|arc-connected]], thus [[Path connectedness|path-connected]] and [[Connectedness|connected]].

## Smooth properties

See [[Infinitesimal calculus MOC]].
We make $\mathbb{R}^n$ into a $C^\alpha$-[[differentiable manifold|manifold]] for any $\alpha$ by taking the maximal atlas induced by the identity chart $\{ 1_{\mathbb{R}^n} \}$.

## Geometric properties

Real coördinate space is commonly considered with the [[affine space|affine]] geometry of [[Euclidean space]].

## Measure properties

Real coördinate space is usually given the [[Lebesgue measure]].

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