I am reading An Introduction to Algebraic Topology by Rotman. After proving

Theorem 2.7: For every $k\geq 0$, euclidean space$\mathbb R^n$ contains $k$ points in general position,

the book remarked: There are other proofs of this theorem using induction on k. The key geometric observation needed is that $\mathbb R^n$ is not the union of only finitely many proper affine subsets . I want to prove that observation.


Every affine proper subspace of $\mathbb{R}^n$ i contained in an affine hyperplane.

Now, let $C=\{(t,t^2,t^3,\ldots,t^n),\,t \in \mathbb{R}^n\}$.

Since every nonzero polynomial with degree at most $n$ has at most $n$ roots, an affine hyperplane of $\mathbb{R}^n$ contains at most $n$ points in $C$.

So if you have a covering of $\mathbb{R}^n$ by proper affine subsets, you need at least $|\mathbb{R}|$ of them.


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