Real coördinate space

Real coördinate space $ℝ 𝑛$ is the set of $𝑛$ -tuples of Real numbers.

Topological properties

Considered as either a metric topology or product topology, $ℝ 𝑛$ has the following properties

Heine-Borel theorem: Subsets compact iff closed and bounded.
$ℝ 𝑛$ is arc-connected, thus path-connected and connected.

Smooth properties

See Infinitesimal calculus MOC. We make $ℝ 𝑛$ into a $𝐶 𝛼$ -manifold for any $𝛼$ by taking the maximal atlas induced by the identity chart ${1 ℝ 𝑛}$ .

Geometric properties

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

Measure properties

Real coördinate space is usually given the Lebesgue measure.

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Table of Contents

Real coördinate space

Topological properties

Smooth properties

Geometric properties

Measure properties

Graph View

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