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

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

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.

Title	$\mathbb{R}^{n}$ is not a countable union of proper vector subspaces
Canonical name	mathbbRnIsNotACountableUnionOfProperVectorSubspaces
Date of creation	2013-03-22 14:59:03
Last modified on	2013-03-22 14:59:03
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Result
Classification	msc 54E52

ℝn is not a countable union of proper vector subspaces

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