PlanetMath: $\mathbb{R}^n$ is not a countable union of proper vector subspaces

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$\mathbb{R}^n$ is not a countable union of proper vector subspaces

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Proof

We know that every finite dimensional proper subspace of a normed space is nowhere dense. Besides, $\mathbb{R}^n$ is a Banach space, so the results follows directly.

" $\mathbb{R}^n$ is not a countable union of proper vector subspaces" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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54E52 (General topology :: Spaces with richer structures :: Baire category, Baire spaces)

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