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Jan 28, 2018 at 14:17 history edited Qfwfq CC BY-SA 3.0
(reformulated Lemma in equivalent way; deleted wrong proposition)
Jan 28, 2018 at 14:12 history edited Qfwfq CC BY-SA 3.0
deleted 66 characters in body
Jan 28, 2018 at 2:11 history edited Qfwfq CC BY-SA 3.0
added 107 characters in body
Jan 28, 2018 at 1:51 history edited Qfwfq CC BY-SA 3.0
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Jan 28, 2018 at 1:43 comment added Qfwfq Clearly my lemma didn't make much sense, as stated. I have now edited it; the linked MSE answer is now (I think) not a counterexample to my lemma. But it is still a counterexample to its corollary (i.e. my "Proposition").
Jan 28, 2018 at 1:39 history edited Qfwfq CC BY-SA 3.0
(lemma stated in a non sound way)
Jan 28, 2018 at 1:11 comment added Arrow Dear Qfwfq, thank you very much for this detailed answer. At first glance I do not notice the local injectivity assumption anywhere. Does this mean the subset $\{(x,y,\sqrt[4]{x^2+y^2})\mid x,y\in \mathbb R\}\subset\mathbb R^3$ is not a counterexample as I claimed in my post? As stated in the link, any vector space containing the $z$-axis would satisfy the limit condition, particularly two distinct planes intersecting at the $z$-axis, in seeming contradiction to the last proposition of your answer.
Jan 28, 2018 at 0:49 history answered Qfwfq CC BY-SA 3.0