I need to construct an example of Jordan path in $\mathbb{C}$ for which it’s support has positive plane Lebesgue measure. And how to prove proof that for closed Jordan path it’s support is homeomorphic to circle (intuitively it’s clear but I want more precise proof)
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1You can close this curve and the homeomorphism is because you have a continuous bijection between compact Hausdorff. – MoonLightSyzygy Dec 25 '19 at 18:01
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I need a curve with positive one-dimensional Lebesgue measure. I think this example is complicated – BeesaFangirl DOTO Dec 25 '19 at 18:04
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1You said 'plane Lebesgue measure'. Just the circle has positive length. – MoonLightSyzygy Dec 25 '19 at 18:06
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It is needed to be support of positive measure – BeesaFangirl DOTO Dec 25 '19 at 18:46
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1I don't know what are saying. The interesting question that sounds like what you are asking is answered by the link above. If not, clarify what you call support and clarify which measure you want. – MoonLightSyzygy Dec 25 '19 at 18:50