Let $A \subset \Bbb{R}^2$ be compact and contractible. I need to show $\Bbb{R}^2 \setminus A$ is connected. I know since $A$ is a compact subset of the plane, it is closed and bounded thus its complement is open. Do I also need to show its closed to show connectivity or could I appeal to a theorem?
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It is not clear what do you want to accomplish with showing that it is closed. – Mariano Suárez-Álvarez Feb 20 '23 at 21:56
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I suppose what i meant to say is that the only clopen subsets of the complement are whole space and empty set which would imply its connected @MarianoSuárez-Álvarez – homosapien Feb 20 '23 at 22:03
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You have to use some algebraic topology tools. Do you know Alexander duality? – Moishe Kohan Feb 20 '23 at 22:34
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I do not know Alexander duality, there must be a simpler way to do it lol @MoisheKohan – homosapien Feb 20 '23 at 22:41
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1OK, what tools of AT do you know? – Moishe Kohan Feb 20 '23 at 22:42
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Just some basic homotopy theory and fundamental group theory. Borsuk Ulam, Borsuks Lemma, Jordan curve theorem, hairy ball theorem. @MoisheKohan – homosapien Feb 20 '23 at 22:43
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How did you prove Jordan curve theorem? – Moishe Kohan Feb 20 '23 at 22:45
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@MoisheKohan by first showing $S^2$ minus the curve has in fact only two components then showing the curve is their common boundary. – homosapien Feb 20 '23 at 22:52
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Ive been hinted to use the Borsuk Lemma but we are in the real plane not the 2 sphere.. so I dont see that connection.. – homosapien Feb 20 '23 at 22:54
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You can translate this to a problem on $S^2$ by noting that since $A$ is compact, under stereographic projection it gives a compact subset of $S^2$. Then you can try to show that the complement in $S^2$ is connected. – kamills Feb 20 '23 at 22:56
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@kamills do I use the nullhomotopy lemma together with the Borsuk Lemma? After converting to the unit 2 sphere ? – homosapien Feb 20 '23 at 23:16
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@kamills what i was thinking was defining $f: A \to S^$ via the identity map sending elements to themselves then applying the Borsuk lemma? – homosapien Feb 20 '23 at 23:36