I'm new to topology and am trying some exercises in Gaal's Point Set Topology (p 63). How do I show that the set of limit points of the product space $(A × B)' = (A' × \overline B\ ) ∪ ( \overline A\ × B')$? Any help will be appreciated!
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What have your tried? – Henno Brandsma Nov 20 '21 at 15:31
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If $A=B={\frac1n\mid n \in \Bbb N} \subseteq \Bbb R$, then $(0,0) \in (A\times B)'$ but it'a not in the union. So a false statement. – Henno Brandsma Nov 20 '21 at 15:52
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I see - thank you. Perhaps it should have read $(A × B)' = (A' × \overline B\ ) ∪ (\overline A\ × B')$? I don't have a deep enough understanding of the material yet and will spend more time on it before attempting exercises. – user604270 Nov 20 '21 at 16:58
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1the one with closures is true yes. – Henno Brandsma Nov 20 '21 at 16:59
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Would you mind giving me a hint as to how to show that the above one with closure is true? I suspect it has to do with the fact that $\overline A\ = A ∪ A'$ and the definition of the product topology, but I am lost... – user604270 Nov 20 '21 at 18:09
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someone else please make it a dupe to this ? I cannot. – Henno Brandsma Nov 21 '21 at 14:40