Answer to Question #213034 in Functional Analysis for smi

Question #213034
Show that R^n and C^n are not compact.


1
Expert's answer
2021-07-05T16:51:47-0400

First of all let us show that Rn\mathbb{R}^n is not compact. There are two possible definitions of compactness for metric spaces (Bolzano-Weierstrass sequence property and Borel-Lebesgue open cover property), let us disprove both of them for Rn\mathbb{R}^n.

  • Let us consider a sequence un=(n0...0)u_n=\begin{pmatrix} n \\0 \\ ... \\ 0 \end{pmatrix} for nNn\in \mathbb{N}. This sequence does not admit a converging subsequence, as for any subsequence the first coordinate diverges to infinity. Therefore, Rn\mathbb{R}^n does not admit the Bolzano-Weierstrass property and thus is not compact (as it is metric).
  • Let us consider an open cover Un=B(0,n)U_n=B(0,n) by the open balls centered at origin with a growing radius. This open cover does not admit a finite subcover, as for any finite subcover we have nNUnB(0,N+1)Rn\bigcup_{n\leq N} U_n \subseteq B(0,N+1)\neq \mathbb{R}^n. Therefore, Rn\mathbb{R}^n does not admit the Borel-Lebesgue property and thus is not compact.

To prove that Cn\mathbb{C}^n is not compact, we can either use the former approach or use a simpler argument : if Cn\mathbb{C}^n were compact, then RnCn\mathbb{R}^n\subseteq \mathbb{C}^n being a closed subset of a compact space, would be compact too. However we have just proven the contrary, and thus Cn\mathbb{C}^n is not compact.


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