Mathematics MOC

Real coördinate space

Real coördinate space is the set of -tuples of Real numbers.

Topological properties

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

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

Smooth properties

See Infinitesimal calculus MOC. We make into a -manifold for any by taking the maximal atlas induced by the identity chart .

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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