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If every Jordan curve $J \subset D$ is null-homotopic in a domain $D \subset \mathbb{R}^2$, then is it clear that every closed curve $C \subset D$ is also null-homotopic in $D$?

Apparently the set $\mathbb{R}^2 \setminus C$ is an union of countably many Jordan domains $D_i$, but it's not clear to me that the result for every $\partial D_i$ is enough. What is the right approach?

EDIT: Before you mofos shut down my question with silly down votes I ask to check out the following link http://faculty.up.edu/wootton/Complex/Chapter8.pdf

There this question is answered with seemingly simple compactness argument. It would be nice if someone of you can explain why those argument are enough?

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Hulkster
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  • @MoisheCohen Possible or not, I don't understand. Visually this problem surely is trivial. – Hulkster Aug 26 '17 at 03:25
  • That answer is a contrapositive to my question, but still there should be a easier proof. – Hulkster Aug 26 '17 at 04:47
  • @MoisheCohen can you check out my edited question? But thanks anyway for your earlier answer. – Hulkster Aug 26 '17 at 07:53
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    This "nice argument" (Lemma 1.5 is fine but seems to have nothing to do with Lemma 1.4) strikes me as meaningless (as is the definition of "interior" given in Lemma 1.4). For instance, what is the "interior" of the Peano curve according to the given definition? I suggest, you read something else instead. There are some really nice books on complex analysis... – Moishe Kohan Aug 26 '17 at 13:06

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HINT: Every closed curve is homotopic to a closed polygon. This one will be easier to handle with your argument.

orangeskid
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  • But if I know the result for simple closed curve, then the case of simple closed polygon is no avail. I still don't know how to shrink $C$ inside $D$. – Hulkster Aug 26 '17 at 03:31
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    @Hulkster: you show that every closed polygon is homotopic to a point. The closed polygon is composed of several closed simple polygons. – orangeskid Aug 26 '17 at 08:08