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Lemma: If A and B are compact subset of $\mathbb{R}$ then $A\times B$ is compact in $\mathbb{R}^2$.

Proof: let V={$V_{\alpha}\subseteq \mathbb{R}^2 |\alpha \in I$} is an open cover of $A\times B$. So, $A\times B \subseteq \bigcup_{\alpha \in A} V_{\alpha}$. $A$ and $B$ $\subseteq A \times B \subseteq \bigcup_{\alpha \in A} V_{\alpha}$, since $A\times${0} and $B\times${0} are equal to $A$ and $B$. Thus, V is also an open cover of $A$ and $B$. Since $A$ and $B$ are compact, there exist some finite subcover of V i.e. $A\subseteq V_{n_1} \bigcup ..... \bigcup V_{n_k}$ and $B \subseteq V_{m_1} \bigcup ..... \bigcup V_{m_l}$, where $k,l \in \mathbb{N}$. Therefore, $A \times B \subseteq V_{n_1} \bigcup ..... \bigcup V_{n_k} \bigcup V_{m_1} \bigcup ..... \bigcup V_{m_l}$. Which is a finite subcover of V.

Is my proof correct and satisfactory? Can this lemma be generalize to metric space?

user264745
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  • An open set in $\mathbb R^{2}$ is not a subset of $\mathbb R$, so $V$ cannot be an open cover for $A$. Also, $A \times {0} \neq A$. – Kavi Rama Murthy Oct 25 '21 at 08:29
  • You are indexing your cover by $A$ but $A$ is already being used for something else. – Aryaman Maithani Oct 25 '21 at 08:35
  • If you are aware of the description of compact subsets of $\Bbb R^n$ (closed + bounded), then you can use that for a much easier proof. – Aryaman Maithani Oct 25 '21 at 08:36
  • About the generalization: the product of any compact topological spaces is always compact. – Berci Oct 25 '21 at 08:36
  • $B$ and $B\times {0}$ are not equal. it would be advantageous to write each $V_{\alpha}$ as $U_{\alpha}\times W_{\alpha}$ for open sets $U_{\alpha}$ and $W_{\alpha}$. – C Squared Oct 25 '21 at 09:58
  • @Berci non-trivial problem when you start producting infinitely many topological spaces – C Squared Oct 25 '21 at 10:00
  • @KaviRamaMurthy I used my 220 IQ. My motivation for that was theorem 1.26. – user264745 Oct 25 '21 at 19:37
  • @CSquared I guess similar answer is given in this https://math.stackexchange.com/a/2018473/861687 ,but I don’t understand some notation and terminology like $U_y(x)$, open box neighbourhood, $N_{x}$ etc. can you please help with that? – user264745 Oct 25 '21 at 19:48

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